for the Eclipse treatment planning system

Clinical Perspectives | Acuros XB Acuros® XB advanced dose calculation for the Eclipse™ treatment planning system Gregory A. Failla1, Todd Wareing1, ...
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Clinical Perspectives | Acuros XB

Acuros® XB advanced dose calculation for the Eclipse™ treatment planning system Gregory A. Failla1, Todd Wareing1, Yves Archambault2, Stephen Thompson2

Introduction

Background

The Acuros XB advanced dose calculation (Acuros XB) algorithm was developed to address two strategic needs of external photon beam treatment planning: accuracy and speed. In external photon beam radiotherapy, heterogeneities introduced by materials such as lung, bone, air, and non-biologic implants may affect patient dose fields, especially in the presence of small or irregular fields. Acuros XB uses a sophisticated technique to solve the Linear Boltzmann transport equation (LBTE) and directly accounts for the effects of these heterogeneities in patient dose calculations. Acuros XB provides comparable accuracy to Monte Carlo methods in treatment planning for the full range of X-ray beams produced by clinical linear accelerators, 4 MV – 25 MV with improved calculation speed and without statistical noise.

The Boltzmann transport equation (BTE) is the governing equation which describes the macroscopic behavior of radiation particles (neutrons, photons, electrons, etc.) as they travel through and interact with matter. The LBTE is the linearized form of the BTE, which assumes that radiation particles only interact with the matter they are passing through, and not with each other, and is valid for conditions without external magnetic fields. For a given volumetric domain of matter, subject to a radiation source, under the above conditions the solution to the LBTE would give an “exact” description of the dose within the domain. However, since closed form solutions (analytic solutions) to the LBTE can only be obtained for a few simplified problems, the LBTE must be solved in an open form, or non-analytic, manner.

Additionally, Acuros XB calculations are minimally sensitive to the number of fields in a plan such that calculation of the dose in a VMAT plan is almost as fast as for a single field. The effect of which is that while single field dose calculations are somewhat slower than with Eclipse Analytical Anisotropic Algorithm (AAA), Acuros XB is faster for VMAT calculations in a single workstation environment.

There are two general approaches to obtaining open form solutions to the LBTE. The first approach is the widely known Monte Carlo method. Monte Carlo methods do not explicitly solve the LBTE; they indirectly obtain the solution to this equation. The second approach is to explicitly solve the LBTE using numerical methods [ref. 1]. Methods used to explicitly solve the LBTE equation, such as those of Acuros XB, are relatively new to the medical physics community.

Acuros XB is fully integrated into the Eclipse distributed calculation framework (DCF) as a new dose calculation algorithm and uses the multiple-source model originally derived for AAA. Therefore, the user will also appreciate that AAA beam data can be imported in the Acuros XB beam model and only requires reconfiguration before it is ready to be used for dose calculations.

Both Monte Carlo and explicit LBTE solution methods such as Acuros XB are “convergent.” That is, with sufficient refinement both approaches will converge on the same solution of the LBTE. The achievable accuracy of both approaches is equivalent and is limited only by uncertainties in the particle interaction data and uncertainties in the problem being analyzed. In practice, neither Monte Carlo nor explicit LBTE solution methods are exact, and both 1Transpire 2Varian

Inc., Gig Harbor, Washington*

Medical Systems, Palo Alto, California

methods produce errors. In Monte Carlo, errors are random and result from simulating a finite number of particles and following each particle as it interacts with a medium. When Monte Carlo methods employ techniques to accelerate solution times or reduce noise, systematic errors may be introduced. In the explicit LBTE solution methods, errors are primarily systematic and result from discretization of the variables in space, angle, and energy. Larger steps in the discretization process result in a faster solution, but less accuracy. In both methods, a trade-off exists between speed and accuracy. Differences between the two methods may also result from the treatment of charged particle Coulomb interactions. Model- or correction-based algorithms such as pencil beam or collapsed cone convolution are only convergent under the exact conditions in which their dose kernels are generated. The impetus behind the development of explicit LBTE solution methods was to provide a rapid alternative to Monte Carlo simulations, which are known to be time intensive. A second benefit of LBTE is the absence of statistical noise. Many of the methods contained within Acuros XB were originally developed in a prototype solver called Attila®, which was co-authored by the founders of Transpire, Inc. while at Los Alamos National Laboratory [ref. 2, 3]. The development of the Acuros XB external photon beam prototype was funded in part through a Small Business Innovation Research (SBIR) Phase II Grant from the National Cancer Institute. Acuros XB in Eclipse – Source Model Acuros XB in Eclipse leverages the existing AAA machinesource model. This model consists of four components: • Primary source – user-defined circular or elliptical source located at the target plane which models the bremsstrahlung photons created in the target that do not interact in the treatment head. • Extra focal source – Gaussian plane source located at the bottom of the flattening filter, which models the photons that result from interactions in the accelerator head outside the target (primary in the flattening filter, primary collimators, and secondary jaws). • Electron contamination – represents the dose deposited in the build-up region not accounted for by the primary and extra-focal source components.

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• Photons scattered from wedge – represents the scatter from hard wedges, where present. Implemented with a dual Gaussian model, where the width of the Gaussian kernel increases with distance from the wedge. A detailed description regarding these sources can be found in the paper on AAA photon dose calculation by Sievinen et al. [ref. 4]. Acuros XB in Eclipse – Patient Transport and Dose Calculation

(The Acuros XB solution methods are briefly described here, with a detailed overview provided in the Appendix.) The Acuros XB patient transport consists of four discrete steps, which are performed in the following order: 1. Transport of source model fluence into the patient 2. Calculation of scattered photon fluence in the patient 3. Calculation of scattered electron fluence in the patient 4. Dose calculation Steps 1 through 3 are performed to calculate the electron fluence in every voxel of the patient. Once the energy dependent electron fluence is solved, the desired dose quantity (dose-to-medium or dose-to-water) is computed in Step 4. Step 1 is the only step repeated for each beam, and Steps 2 through 4 are performed once, regardless of the number of beams. In the case of VMAT, each beam will have a large number of orientations, and Step 1 is repeated at each orientation and Steps 2 through 4 are performed just once. In Step 1, the machine sources are modeled as external sources and ray tracing is performed to calculate the uncollided photon and electron fluence distributions in the patient. In Steps 2 and 3, Acuros XB discretizes in space, angle, and energy, and iteratively solves the LBTE. In Step 4, the dose in any voxel of the problem is obtained through applying an energy dependent fluence-to-dose response function to the local energy dependent electron fluence in that voxel. Acuros XB supports two dose reporting options: dose-to-water (DW) and dose-to-medium (DM). When DM is calculated, the energy dependent response function is based on the material properties of that voxel. When DW is calculated, the energy dependent fluence-to-dose response function is based on water.

Clinical Perspectives | Acuros XB

Therefore, to calculate dose, Acuros XB must have a material map of the imaged patient. Unlike convolution/superposition algorithms, where heterogeneities are generally handled as density-based corrections applied to dose kernels calculated in water, Acuros XB explicitly models the physical interaction of radiation with matter. To do this accurately, Acuros XB requires the chemical composition of each material in which particles are transported through, not only the density. To enable this, Eclipse provides Acuros XB with a mass density and material type in each voxel of the image grid. The Acuros XB material library includes five biologic materials (lung, adipose tissue, muscle, cartilage, and bone) and 16 non-biologic materials, with a maximum supported density of 8.0 g/cc (steel). In Figure 1, an illustration of the differences between DW and DM are presented for a 5 x 5 cm2 6 MV field on a water-bonelung slab phantom. Also shown are results for scaled water density, in which the entire phantom was assigned water material, with varying density according to the region. As shown, DW and DM are identical in water voxels upstream of the bone, and are nearly identical in the lung downstream of the bone. This is expected since in both cases, the electron transport field is identical, and only the electron energy deposition interaction is different. However, for scaled water density there are differences in the build-up region before the bone, in the bone, and in the lung downstream of the bone. These differences highlight the significance of using the actual material composition as opposed to scaling the density of water material.

In Figure 2, differences between DW and DM are presented for a 5 x 5 cm2 18 MV field for the biologic materials in Acuros XB.

Figure 2. Acuros XB depth dose curves for dose-to-water (DW) and dose-to-medium (DM) for a 5x5 cm2 18 MV beam on a slab phantom containing: water (1.0 g/cc), cartilage (1.1 g/cc), bone (1.85 g/cc), lung (0.26 g/cc), adipose tissue (0.92 g/cc), and muscle (1.05 g/cc)

Comparison with Monte Carlo Since Monte Carlo methods are well known in the radiotherapy community, a useful way to understand the methods in Acuros XB is to highlight where and why differences between Acuros XB and Monte Carlo can occur, which are discussed below. Dose-to-water and dose-to-medium Both Acuros XB and Monte Carlo methods calculate DM based on energy deposition, and as shown in the included figures, produce very similar results. However, when calculating DW in non-water materials, Acuros XB and Monte Carlo methods employ different approaches.

Figure 1. Acuros XB depth dose comparison between different dose reporting modes for a 5 x 5 cm2 6 MV field on a waterbone-lung slab phantom. For scaled water density, the entire phantom consisted of water material, but with the density scaled in each region (1.85 g/cc in bone region, and 0.26 g/cc in lung).

Acuros XB calculates the energy dependent electron fluence using the material compositions of the patient, regardless of whether DW or DM is selected. When DW is selected, in non-water materials this is analogous to calculating the dose received by a volume of water which has a minimal impact on the energy dependent electron fluence. Due to the very short range of low energy electrons, this volume may be much smaller than either the dose grid voxel size or detectors used to experimentally measure DW. This effect is most noticeable for bone and non-biologic, high density materials such as aluminum, titanium, and steel. In such cases, when comparing Acuros XB to experimental measurements of DW, it is recommended to explicitly model a small water volume representing the detector in Acuros XB. 3

Monte Carlo methods will generally calculate DM, and employ stopping power ratios to convert DM to DW [ref. 5]. To illustrate the expected differences between the approaches of Acuros XB and Monte Carlo in calculating DW, Figure 3 shows a comparison between energy deposition ratios (water/medium) [ref. 6] and collisional stopping power ratios (water/ medium) [ref. 7] in different biologic materials as a function of electron energy. The energy deposition ratios (Figure 3 left) show the ratio of DW/DM which would be calculated by Acuros XB, and the collisional stopping power ratios (Figure 3 - right) show the ratio of DW/DM which would be calculated by Monte Carlo methods.

Implementation differences As discussed earlier, Acuros XB and Monte Carlo methods are unique in radiotherapy in that both methods explicitly solve for the electron fluence, without the use of pre-calculated dose kernels; however, neither method is exact and in practice differences will occur. A simple way to understand the different approaches between Acuros XB and Monte Carlo is as follows: Analog Monte Carlo methods simulate a finite number of particles, and stochastic errors result from a finite number of particles being tracked. Acuros XB simulates an infinite number of particles, and systematic errors are introduced by discretization in space, angle, and energy. In Acuros XB, the discretization settings are specified internally to provide an optimal balance of speed and accuracy for patient treatment planning conditions. This is analogous to a Monte Carlo code which internally sets a limit on the statistical uncertainty. Acuros XB Calculation Options

Figure 3. (left) Energy deposition ratios (water/medium) and (right) collisional stopping power ratios (water/medium), as a function of electron energy (MeV). The ratio between doseto-water and dose-to-medium in Acuros XB is reflected in the energy deposition ratios; and for Monte Carlo in the collisional stopping power ratios.

Although Acuros XB and Monte Carlo use different methods, calculating DW in a non-water medium is a theoretical quantity, and therefore neither approach is correct or incorrect. Electron cutoff energy Acuros XB employs an electron cutoff of 500 keV (kinetic energy, not including electron rest mass energy). Electrons passing below this energy are assumed to dump all of their energy in the voxel in which they are located. When comparing Acuros XB and Monte Carlo in voxels containing very low density lung or air, the choice of electron cutoff energy may result in differences between the two solvers. However, such differences will generally be isolated to the low density voxels, and will have minor influence on the dose in adjacent tissue.

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The Acuros XB implementation in Eclipse is very similar to that of AAA. A few key points related to the Acuros XB implementation are summarized below, and several differences with AAA are highlighted. Calculation grid voxel size: The Acuros XB calculation grid voxel size can range from 1 to 3 mm. AAA currently supports a voxel size range between 1 and 5 mm. Dose reporting mode: In Acuros XB, DM or DW can be selected as dose reporting options. This concept does not exist in AAA. Plan dose calculation: This is a unique option for Acuros XB. In Acuros XB, the calculation time has a very weak dependence on the number of fields, since the majority of the calculation time is spent calculating the scattered photon and electron fluence, which is performed once for all fields in the plan. When a separate Acuros XB calculation is performed for each field, the scatter calculation phase has to run for every field, which increases the calculation time. Since field weights cannot be edited when plan dose calculation is selected, this option is well suited for rapidly calculating intensity-modulated radiation therapy (IMRT) and Varian RapidArc® radiotherapy technology plans. However, in 3D conformal planning where field weights may be individually changed during optimization, plan dose calculation would generally be turned off.

Clinical Perspectives | Acuros XB

Material specification: Material determination is done in two ways for Acuros XB. The default method used to determine the material composition of a given voxel in a 3D image is based on the HU value. The HU value in the voxel is converted to mass density using the CT calibration curve. This curve can be configured by the users for their specific CT scanner. Once mass density is known in a voxel, the material is determined based on a hard coded look up table stored in the Varian system database. This automatic conversion is used for all voxels with mass density below 3.0g/cc. Any voxel with density higher than 3.0g/cc requires user assignment. Furthermore, the automatic material assignment only assigns biological materials to voxels. Based on their mass density, voxels will be assigned lung, adipose tissue, muscle, cartilage, or bone. Even very low density regions are automatically assigned a material, either lung or air. Users have the option to manually override the automatic material assignment. Configuration: Since Acuros XB uses the same source model as AAA, no additional beam data is needed and the AAA configured data can be imported into the Acuros XB model directly. AAA beam data imported in Acuros XB will need to be reconfigured and all configuration steps will need to be run again to optimize the source model for Acuros XB. The preconfigured beam data available for AAA is also available for Acuros XB. For every DCF version, preconfigured beam data for AAA and Acuros XB is available.

Figure 4. Depth dose comparison (dose-to-medium) between Acuros XB and MCNPX for a 6X 10 x 10 cm2 field on a multi-material slab phantom. Slab materials are as follows: (1) Polystyrene – 1.05 g/cc, (2) Epoxy – 1.04 g/cc, (3) Aluminum – 2.7 g/cc, (4) PMMA – 1.19 g/cc, (5) Titanium alloy – 4.42 g/cc, (6) Radel – 1.30 g/ cc, (7) Wood – 0.70 g/cc, (8) PEEK – 1.31 g/cc, (9) PVC – 1.38 g/cc, (10) Acetal – 1.42 g/cc, (11) PVDF – 1.77 g/cc, (12) PTFE – 2.20 g/cc.

Acuros XB Validation Examples A brief sampling of Acuros XB validation cases with heterogeneities are provided below. Note that in order to fully validate Acuros XB against Monte Carlo N-Particle eXtended (MCNPX), the MCNPX computations were run with a very large number of particles to create results that were very smooth and without statistical uncertainties that may have influenced the validations of Acuros XB. In practice typical Monte Carlo results are much less smooth and statistical uncertainties are clearly visible. Additional validation results can be found in the literature [ref. 8].

Figure 5. Depth dose comparison (dose-to-medium) between Acuros XB and MCNPX for a 20X 10x10 cm2 field on a multi-material slab phantom. Slab materials the same as in Figure 4.

The Acuros XB material library includes 13 non-biologic materials. Figure 4 and Figure 5 compare DM results from Acuros XB and MCNPX on a slab phantom containing 12 of the 13 non-biologic materials for 6 MV and 20 MV.

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The highest density material supported in Acuros XB is stainless steel, with a maximum density of 8.0 g/cc. Figures 6 through 8 present an extreme case where a 2 x 2 x 2 cm3 steel implant (8.0 g/cc) is placed in a water phantom inside an 18 MV 10 x 10 cm2 field. As shown, both codes are in close agreement, even in the high gradient electron disequilibrium regions surrounding the implant. Figures 9 through 11 present comparisons between Acuros XB and Monte Carlo for a half cork (0.19 g/cc) phantom for 5 x 5 cm2 fields and 6 MV and 15 MV beam energies. Figure 6. Phantom containing a 2 x 2 x 2 cm3 8.0 g/cc steel implant steel implant used in Figures 7 and 8. Acuros XB dose contours shown (dose-to-medium) for an18 MV 10 x 10 cm2 field.

Figure 9. Phantom containing a half cork slab (.193 g/cc) used in Figures 10 through 13. Acuros XB dose contours shown (dose-tomedium) for a 6 MV 5x 5 cm2 field.

Figure 7. Depth dose comparison between Acuros XB and MCNPX for an 18 MV 10 x 10 cm2 field impinging on the steel insert phantom shown in Figure 1. Dose-to-medium shown in both codes with dose normalized to 100% at depth of 4.875 cm.

Figure 10. Depth dose comparison between Acuros XB and MCNPX for a 5 x 5 cm2 15 MV field on the half cork slab phantom

Figure 8. Lateral depth dose comparison (depth of 4.875 cm) between Acuros XB and MCNPX for an 18 MV 10 x 10 cm2 field impinging on the steel insert phantom shown in Figure 1. Doseto-medium shown in both codes with dose fields normalized to 100% at centerline depth of 4.875 cm.

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Figure 11. Lateral dose comparison between Acuros XB and MCNPX for case shown in Figure 10, at depths of 4.625, 17.875, and 21.125 cm.

Clinical Perspectives | Acuros XB

Figure 13 shows the result of an Acuros XB RapidArc dose comparison with the Radiological Physics Center (RPC) head and neck phantom. TLD measurements are within 2% of calculated dose for the 3 mm x 3 mm calculation grid size.

Figure 12 presents comparisons between Acuros XB and Monte Carlo for 2 x 2 cm2 6 MV fields on a water phantom containing a 2 x 2 x 10 cm3 block of air, which simulates an esophagus.

Figure 12. Depth dose comparison between Acuros XB and MCNPX for a 2 x 2 cm2 6 MV field on a phantom containing a 2 x 2 x 10 cm3 air block representing an esophagus. Electron energy cutoff for both Monte Carlo and Acuros XB is 500 keV.

Figure 13. Acuros XB RapidArc plan for Radiological Physics Center (RPC) Head and Neck Phantom (figure courtesy of Firas Mourtada, Ph.D., UT MD Anderson Cancer Center).

Acuros XB, Heterogeneity “on”; Dose to medium

Measured dose (cGy) TLD position Treatment 1

Treatment 2

Treatment 3

Average

Calculated dose (cGy)

% Error

TLD_54_I

621.4

621.9

621.6

621.6

593.3

4.56%

TLD_54_S

591.2

603.5

608

600.9

591

1.65%

TLD_66_Iant

745.3

742.4

735.5

741.1

734.6

0.87%

TLD_66_Ipost

735.3

744

751.4

743.6

739.9

0.49%

TLD_66_Sant

723.9

736.9

736.5

732.4

726.2

0.85%

TLD_66_Spost

728

733.7

738.4

733.4

726.4

0.95%

TLD_CORD_I

355.3

360.9

362.1

359.4

349.7

2.71%

TLD_CORD_S

357.3

357.2

357.6

357.4

346.4

3.07%

Averaged percentage error (%)

1.89%

Table 1. Measured (TLD) vs calculated doses for RapidArc plan shown in Figure 13 above. Measurement is within 2% of calculation averaged over all TLD locations (courtesy of Firas Mourtada, Ph.D., UT MD Anderson Cancer Center).

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Acuros XB Calculation Times* Calculations of a single or few fields are longer with Acuros XB than AAA. For a 10 x 10 cm2 6 MV field on a 30 x 30 x 30 cm3 water phantom, Acuros XB will calculate the dose on a 2.5 mm voxel grid in about 51 seconds (Dell T5600 with dual six-core Xeon 2.00 GHz processors and 32 GB DDR3 RAM). AAA will require approximately 5 seconds for a similar case. For a 5 x 5 cm2 field on the same phantom, Acuros XB will require about 24 seconds. Larger fields and higher energies take longer to calculate, as do phantoms containing large amounts of bone. Most of the Acuros XB calculation time is in solving for the scattered photon and electron fluencies, which are performed only once for all beams in the plan. As a result, the relative calculation speed of Acuros XB increases with increasing numbers of fields in the plan. For cases with larger numbers of fields, i.e., RapidArc, Acuros XB exploits spatial adaption to speed up calculations in low dose, low gradient regions.

Figure 14. Acuros XB dose field (dose-to-medium) for a 6 MV RapidArc lung case. Total dose calculation time, including source model and patient transport, on a 2.5 mm voxel grid: 69 seconds (178 control points).

Acuros XB becomes faster than AAA for cases with a large number of fields, i.e., RapidArc. As an example, the calculation times for two RapidArc cases are provided in Table 2 below, with screenshots of the Acuros XB dose calculation for the lung and head/neck cases shown in Figures 14 and 15. It should be noted that when increased processing resources are available and utilized, AAA calculation times will decrease given that the calculation times scale linearly with the number of fields. Therefore, the more resources available, as with a FAS environment, the greater the decrease in calculation time. Ultimately, this may result in shorter calculation times for AAA compared to Acuros XB, even for plans with a large number of fields, i.e., the number of control points/ fields in a RapidArc plan. The Monte Carlo equivalent accuracy achieved with Acuros XB should be taken in consideration when encountering this scenario.

Cases

Figure 15. Acuros XB dose field (dose-to-medium) from a 6 MV RapidArc head and neck case. Total dose calculation time, including source model and patient transport, on a 2.5 mm voxel grid: 142 seconds (356 control points).

Acuros XB

Acuros FAS°

AAA

AAA FAS°

Lung (178 control points, Fig. 14)

1 min 9 s

0 min 45 s

1 min 52 s

1 min 15 s

Head & Neck (356 control points, Fig. 15)

2 min 22 s

1 min 30 s

5 min 44 s

1 min 58 s

Table 2. Acuros XB and AAA calculation times shown for representative RapidArc cases. All times shown on a Dell T5600 (Hyperthreading off for Acuros XB) with 2.5 mm voxel grids. Calculation times include both source model and patient transport components.

°Framework Agent Server (FAS)

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Clinical Perspectives | Acuros XB

Conclusion The Acuros XB advanced dose calculation algorithm was developed and implemented in Eclipse to address the accuracy and speed requirement for modern techniques in radiation therapy including IMRT and RapidArc. Acuros XB provides comparable accuracy in treatment planning conditions to benchmarked Monte Carlo methods for the full range of X-ray beams produced by clinical linear accelerators, 4 MV – 25 MV. Validation has been performed to assure dose calculation accuracy in typical and challenging phantom and patient geometries with excellent results. References 1. Lewis EE, Miller WF, “Computational methods of neutron transport”, Wiley, New York, 1984. 2. Wareing TA, McGhee JM, Morel JE, Pautz SD, “Discontinuous Finite Element Sn Methods on ThreeDimensional Unstructured Grids”, Nucl. Sci. Engr., Volume 138, Number 2, July 2001. 3. Wareing TA, Morel JE, McGhee JM, “Coupled ElectronPhoton Transport Methods on 3-D Unstructured Grids”, Trans Am. Nucl. Soc., Washington D.C., Vol 83, 2000. 4. Sievinen J, Ulmer W, Kaissl W. AAA photon dose calculation model in Eclipse. Palo Alto (CA): Varian Medical Systems; 2005. [RAD 7170B] 5. Siebers JV, Keall PJ, Nahum AE, and Mohan R, “Converting absorbed dose to medium to absorbed dose to water for Monte Carlo based photon beam dose calculations”, Phys. Med. Biol. 45 (2000) 983-995. 6. Lorence L, Morel J, and Valdez G, “Physics Guide to CEPXS: A Multigroup Coupled Electron-Photon Cross Section Generating Code,” SAND89-1685, Sandia National Laboratory, 1989. 7. http://www.nist.gov/physlab/data/star/index.cfm 8. Vassiliev ON, Wareing TA, McGhee J, Failla G, “Validation of a new grid-based Boltzmann equation solver for dose calculation in radiotherapy with photon beams”, Phys. Med. Biol. 55(3) 2010. http://iopscience.iop.org/00319155/55/3/002/

*Transpire, Inc. assets acquired by Varian Medical Systems on April 5, 2014 **Eclipse software version 13.5MR1 used for all dose calculations

Appendix Acuros XB solution methods – patient transport The Acuros XB patient transport consists of four discrete steps, which are performed in the following order: 1. Transport of source model fluence into the patient. 2. Calculation of scattered photon fluence in the patient. 3. Calculation of scattered electron fluence in the patient. 4. Dose calculation Steps 1 through 3 are performed to calculate the electron fluence in every voxel of the patient. Once the energy dependent electron fluence is solved for, the desired dose quantity (dose-to-medium or dose-to-water) is computed in Step 4. Step 1 is the only step repeated for each field orientation, and Steps 2 through 4 are performed once, regardless of the number of orientations. Material specification Prior to initiating Step 1, Acuros XB must have a material map of the imaged patient. Unlike AAA, where heterogeneities are generally handled as density-based corrections applied to dose kernels calculated in water, Acuros XB explicitly models the physical interaction of radiation with matter. To do this accurately, Acuros XB requires the chemical composition of each material in which particles are transported through, not only the density. To enable this, Eclipse provides Acuros XB with a mass density and material type in each voxel of the image grid. The Acuros XB material library includes five biologic materials (lung, adipose tissue, muscle, cartilage, and bone) and 16 non-biologic materials, with a maximum supported density of 8.0 g/cc (steel). The fundamental material data used by Acuros XB are known as macroscopic atomic cross sections. A macroscopic cross section is the probability that a particular reaction will occur per unit path length of particle travel, so it has units of cm-1. The cross sections also describe the angular and energy behavior probabilities associated with any given interaction. Macroscopic cross sections are composed from two values: the microscopic cross section for a given reaction (generally given in barns/atom = 10-24 cm2/atom and symbolized by σ~) ~ mass density ρ σ~ and the of the material σ~ σ ρ 3). The expression for the macroscopic σ ρ in g/cm (ρ , given cross σ σsection, σ , is: N ρ σ = a σ~

Naρ ~ N ρN ρ σ σ =σ =a σ~a σσ~ = M

M

M M

where M = Mass of the atom in atomic mass units (AMU) Eq.Eq. 20 20 Eq. 20 Na = Avogadro’s number

Eq. 20

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qˆ q (r , E , Ω qq yey) q

ye

y y

ˆq) q ye (r , E , Ω q →

ˆ) q y (E, Ω

qy

qe



rp

e e

q q

ˆ) q e (E, Ω

qe



σ ty

rp Acuros XB uses coupled photon-electron cross sections proσσteyty = σ duced by CEPXS [ref. 4]. For photon interactions, CEPXS inσ te →σ tet y(also known as incoherent scatter), σ y ( rσ cludes Compton scatter , E) t σt SσRtt e → the photo-electric effect,e and pair production. CEPXS does σ t (σr ,t E= ) σ S not account for Rayleight scatter (also known as coherent→ S RR = σ t (r , E ) t scatter), the effect of σ which is insignificant for dose distributions at energies typical S R in photon beam radiotherapies.

q ye qy qy qe

rp → →p

r p rp

qe

σ



y t



rp → →p

r p rp



rp →

rp →

rp →

ˆ Ω ˆ q) yyeye (→r , ˆE , ye q y (E, Ω ˆ) p ˆ )()r→, E , Ω q ((Er ,Ω Eq), Ω p ˆ q y (E, Ω ˆ) p ˆ q))y ( E , Ω q y (E, Ω ˆ )e q e (E, Ω ˆ) q (E, Ω e ˆ q (E, Ω ˆ) ˆ q))e ( E , Ω q e (E, Ω

rp

p p

p

p

p p

p



σ ty (r , Eσ) y (r→, E ) →

t

e → σσtyyt ((→rr,,EE)σ)y e (→r→, E ) , Macroscopic electron total cross σ t(r , E ) σ σsection, → σ t (er ,→E ) t -1 σ t ( r , E )σσ(etr→(,→rE, )E ) e → units of cm σ σt t (r , Eσ) t (r , E→) σ tte → S R (r , E ) → (→r , E ) → σ t SR Macroscopic total cross section, σ t ( r , E )σ t ( r , E ), units of→cmS-1R (r , E ) σ t S (→r , E ) → S RR(r , ES) R (r , E ) SR

ey tt

Restricted collisional plus radiative stopping power, →

S R (r , E )

The first term on the left hand side of Equations 1 and 2 is the streaming operator. The second term on the left hand In Steps 1 through 3, Acuros XB solves the time-independent side of Equations 1 and 2 is the collision or removal operator. three-dimensional system of coupled Boltzmann transport Equation 2 is the Boltzmann Fokker-Planck transport equaequations (LBTE) shown below (for brevity the dependent tion, which is solved for the electron transport. In Equation 2, variables have been suppressed in the equations): the third term on the left represents the continuous slowing down (CSD) operator, which accounts for Coulomb ‘soft’ Eq. 1 electron collisions. The right hand side of Equations 1 and 2 → include the scattering, production, and the external source γ γ γ γγ γ ˆ → Eq. γ 1 γγ e γ Ω ⋅ ∇Ψ →+ σ→→t Ψγ = qγ →+γ q ,,γγ ˆ γ γ γ γ qe+ q1 , qterms and from q e Eq. ˆ ⋅∇ γ q + γ q⋅ γ∇,Ψ γγ+ σ γ Ψ γqγγ and γ Ψ γ +γ σˆ γγ Ψ γγ →γ = and qqee). Ω = q Eq. the1AAA source module (qqγ and and ˆ ⋅Ω γγ γ Eq. 1 →→ t q and q Eq. 1 Ω ⋅ ∇ Ψ + Ψ = q + q , σ t Ω ∇ Ψ + Ψ = q + q , σ γ γ γ γγ γ ˆ γ +t σ γΨ γ γ = q γγγγ + γ, tΨ ˆ γ e Eq. 1 ˆ Ω ⋅ ∇ Ψ q γ γ γ Eq. 1 Eq. 1 Ω ⋅ ∇ Ψ + = q + q , σ ˆΩ⋅ ⋅∇∇ΨΨ ++σσt tΨΨ ==qq t++qq , , q and q Eq. 1 Ω γ γ γγ t Eq. 1 → + σ tγ Ψ + qe2γ ,e ∂→ Eq. Eq. 2 e e γe → ee γ scattering ˆ ⋅ ∇=Ψqe + The →+ q γ + q →Ψ − γ γ γγ γ Ω S Ψ = q , σ q q e Eq. 2and production sources are defined by: ∂eeeq , γe e Eq. γ γγ γe = eγq →e γR ∂ˆ γ⋅Ψ t e∂ Ψ eand ˆe σ∂⋅ˆ∇ 21 + q γe Eq. ee Ψ e+ ee 1 ∇ σ ˆΨ →e→ e Ω ˆ→ ⋅ ∇ Eq. Eq. 2 e Ω eΩ e= eq ee + q q σ Eq. 2 eΨ eet+ e,qe e − e+ γΨ →e ∂ ⋅ ∇ Ψ S Ψ = + q , σ ∂ E Ω Ψ + Ψ − S = q + + q , ˆ t Eq. 2 ∂ e e e ee e e → +σ Ω Ψ − q+eeSeeqRt+Ψq+eγγeγqe+ = ,qeee e, +Rqγe +Eq. qe , 22 Re = Ω ⋅Ω ∇ Ψ −eΨ S∂σσR Ψ Eq. 2 eS ˆˆΩ e t σ e∇ e−e + ∂E t→ ˆσtte→⋅⋅Ψ ˆΨ ⋅⋅ ⋅∇ Ω ∇ Ψ −e eq∂= , ΨΨ SRRΨ +qq γγ=++∂qqqE q Eq. ,, q 3 + qEq. ∂ tSEq. Eq. 33 E−∂∂+E Ω ∇∇Ψ ΨΨe + +γˆ+eσ ΨΨ =E=Eqγ qS Rγγ+Ψ , γ+ γ 2 γγ Eq.33 Eq. ˆEγ⋅Ψ t tΨe γ∂− R γΨ Eq. Eq. 1 Eq. 3 ∂ Ω ∇ + Ψ = q + q , σ ∂ − S R Ψ e = q →ee + qΩ qΨ , Eq. 1 ⋅+∇ + = q + q , σ t ∂Et ∞ → ∞ ∞∞ 3 Eq. 2 → → e ∂ e e e ∂ ee eγe ee e 2 γe ∂E ∞ → → Eq. e e e e ˆ γγ γγ γ → γ → ˆwhere ˆ ) = dE ʹq γγd(Ωr ʹ, σ ˆΩ⋅ʹΩ ˆ ʹγγ)Ψ ˆγγγγ,ʹΩ Ωt→⋅Ψ ∇Ψ− +→σ tSΨ ˆ(→)r ,=E∞ʹ d→ ˆ→r→⋅,,Ω ˆ ʹ,)ˆΩ → ʹ,qqΩ Ω ⋅ ∇Ψ + σ Ψ− γγ= q S Rγ+Ψq =+ q , + q + q , q r , E , Ω E , Ω r , E ) ˆ ʹ ʹ E , Ω E d ( r → E Ψ ( r ,ddEEE σ γγ (→ γγ γ ,(E R s γ γ ( E ) = ˆ ˆ ˆ ˆ →γ s ( r , E , Ω ) = ˆ γγ γ Eγ q (r , E , Ω) = dE ʹ γγdΩ→ʹ σ s ˆ(r , E ʹ → E , Ω ⋅ Ωʹ)γγΨ→ (r , E ʹ, Ωʹ )ˆ ˆ Eq. 1 Eq.Eq. Et Ψ Ω ⋅ ∇Ψ γγ +=σqγγ∂Ψ → Ω γ → ∇ ˆˆ+e ⋅∂⋅σ → ∂ +γ q== qq,γγeγe++ qqγ eee,, 0 4π ( r , E , Ω) = 0 dE ʹ4π dΩ ʹ σ Eq.Eq. 11 3 Ω ΨΨ ʹ ʹ 2 e +e σ e 2 γe ∂∇ q ( r , E → E , Ω ⋅ Ω ) Ψ ( r00, E Ψγ ˆ Eq. e e⋅ ∇eΨ e tt Ψ ee s ∞ γ ˆ 0 4 π γ Ω + Ψ − S Ψ = q + q + q , σ → q Ω = + if Eq. , E time γ tSΨ R+ q γΨ⋅ γ∇Ψ + σ t Ψ y− → → → → RΨ γ 4 ˆ Eq. 4 = Angular photon fluence (or fluence not γγ γγ γ r = ( x , y , z ) Ψ 0 4 π , Ψ γ → → ∂ E Eq. 4 Ψ ( r , E , Ω ) γ ˆ ˆ ˆ → Eq. 1 → q , Eq. 4z ) Ψ Eq. y4 → q (r , E , Ω) = ∞ dEq. E ʹ 4dΩʹ σ s (r , E ʹ → E , Ω ⋅ Ωʹ)Ψ (r , E γ ∂→Eγy →γy → γγ ˆ → γy → Ψ ˆ Ψγ → e Ω E E r = ( x , y , ˆ ∞ →→= ( x r , y , z ) , Ψ ( r , E , Ω ) Ψ , → → ˆ ∂ E Eq. 1 Ψ ( r , E , Ω ) ˆ → E ⋅ ∇ Ψ + = q + q , r = ( x , y , z ) σ → y r = ( x , y , z ) γer ,yE γ,ˆas ,e → ee yaeγγ → ∞∞ (function r→r,+γe= E ,(γΩ →e ΨeeΨ ,∂eγΩΨ=,)Ω tΨ ˆ ⋅ ∇Ψ integrated), Eq.Eq. ee ˆ−⋅ ∇ ))+)=qe(e x,,,E → ˆ(xxqx,+),,)e,yy,qof γeˆ → 0 γ → 1E ˆΨ 4π ˆ =)Ψ Ω ΨΩ Se, + σ y,, 2 yζ γ → Eq. 22q,γγ4 y,,γγ,ezzerzposition, eˆΩ ˆ ) = ∞ dE ʹq γde (Ωr→,ʹ σ ˆ ʹγ)eΨ ˆγeγ,ʹeΩ ,z ) Eq. Ψ ˆqΩ ,Ψ )(rr→r∂,,t ,E Ω ∇tΨ Ψ +=σσ( μ Ψ −((Ψ Ψ =, E=qq=q,+eeee η(ˆΨ E,SS,Ω ) qe+(rqrr= EE ˆ(→r→⋅,,Ω ˆ ʹ,)ˆΩ ˆˆ+⋅⋅σ∇ R → → → (Ω ʹ,qΩ r , E , Ω ( r , E → E , Ω ⋅ Ω r , E ) ˆ ʹ ʹ ʹ E tΨ R) E ) = E d Ω ( r → E Ψ ( r ,ddEEE σ e (→ γ,seΩ γ , (E Ω + − + q + q , ∞ʹ d y → γ ˆ q E ) = ˆ ˆ ˆ ˆ s t R ˆ γ ∂ E ( r , E , Ω ) = E r =Ψ ( x, y , z ),ˆenergy, ˆ direction, q 4(r , E , Ω) = dE ʹ γde Ω→ʹ σ s ˆ(r , E ʹ → E , Ω ⋅ Ωʹ)γeΨ→ (r , E ʹ, Ωʹ )ˆ ˆ = (,∂μ ηand Ψ e (re , E ,ΨΩ) Eq. → Eq. Ω = ( μ ,η , ζ )Ω =→ ( μ ,η , ζ ) =Ω ,,γγ,ζη ),, ζζΩ EEE, ˆˆΩ 2 → γ e →Ω ee 0 4π ( r , E , Ω) = 0 dE ʹ4π dΩ ʹ σ ( r , E ʹ → E , Ω ⋅ Ω ʹ) Ψ γ ( r0, E ˆζ))γ)= ˆ(ˆγy=μ=e,∂(→η(μ q γΩ , , ) μ η ζ y (→→ → , , μ η → 0 + q γeΨ + q , ˆ ˆ s E ∂ ∞ 0 4 π Ω = ( , , ) r = ( x , y , z ) μ η ζ ˆ e 1 E , Ψ ( r , E , Ω ) 2 e Ω→ eΨ e + eq ee ,(γγrr,γeE e y , ze) → Eq. ⋅ ∇ Ψ = + q σ Ψ , Ω ) = ( x , , γ γ γ γ e → ˆ Ψ ( r , E , Ω ) ˆ e t Ψ ˆ−⋅ ∇ e+Ψ ∂→,t E γe Ω Ψ,⋅eζγ∇ =, Ω q, Eq. ,Ψ Ωyy= (Ψ )Ψ ˆ + σΨ μ ,Ω η ˆ ) Eq. Eq.Eq. ˆqΨ 21 5 e Ψ eS→R ee e, E , Ω γe 5 γe → γ → Ψ =eqe+(=→rq→,+ q+, Ω σ eΨ 0 Eq. 45 π5 ʹ γe → t eΨ Eq. ( r ˆ ˆ ˆ e= ˆ ( r ) ˆ Ψ e → ʹ ʹ ʹ Ψ → Eq. 5 Ω ⋅ ∇ Ψ + Ψ − S q + q + q , e Angular electron fluence, σ Eq. 5 q ( r , E , Ω ) = d E d Ω ( r , E → E , Ω ⋅ Ω ) Ψ ( r ,E σ Ψ E ) → Ψ ( r , E , Ω ) e ˆ Ψ e( r yy, E → E ΨˆΨ q Rγˆ)ˆ ˆy γγ s ˆ,ˆγζ) ) → ∞ ,,,,Ω ΨΨyy ∞ ˆ ⋅∂t ∇ EE ,,q(Ω r,μ,E Ω ))Ω ∞ Ω =(()rr→(r→→,μ qyy (tγ,E rΨ Eˆ ,ee5 → ∞ 1 → Eq. Ωγqey=+ ,ˆˆE η ζΨ yy , y,, zyy) Eq. Ω Ψ +,η ,yyy, z r)→ = ( xE σ Ψ ((∂(rr→(E Ω )),= ∞ Ψ , Ω ) → → → e → yyq yy yyq = x , eeˆ → 0 e → 4π ˆ → → ˆ Ψ , E , Ω ee ee e ˆ ˆ ˆ ˆ ˆ → q yy q → ee yy ˆ q ( r , E , Ω ) yy ˆ → Ψ (r , Eˆq,qΩ q yy,(Er→→→γ,,eΩ Eˆ ,)Ω qee (→r , E , Ω r ,ʹ E )(r,⋅,Ω ˆ ) =(r ,ddEEE γ) ∂ Eee ,s Ω(→r) ,=E∞ʹ d→ E ʹE ,dΩ (re ,(E →ʹ, Ω Ψ Ω Eq. e e e ee ˆqˆeq)),)yy ((r→r,,EE,,Ω qE Eʹ,)ˆΩ ˆ ) = dE ʹq d(Ωr ,ʹ σ ˆΩ⋅ʹΩ ˆσ ʹ)Ψ ˆee,ʹ Ω e Ψ e + γ→ ˆ ) 2 Ψ⋅yy∇ qqˆ(qryy (q→(rr→r,, ,E , ,Ω yy ∂→→e =qqˆ Ω e source, ΩqΨ Ψe − e eSˆ ReΨ = yyEq. Photon-to-photon scattering Ψ1σqγΨ E 2 q 5 (r , E , Ω) = 0 dE ʹ 4πeedΩ→ʹ σ s ˆ(r , E ʹ0 → E4π, Ω ⋅ Ω ʹs)eeΨ→ (r , E ʹ,qΩ ʹ ()ˆr , Eˆ, Ω) = e+ ee e → ˆ→Ψ Ω q γ , q ee ,+ ,qˆ+ˆxΩ η,,(Ω ζ),)q)y→γ→,e z+) q e , )E, , Eq.Eq. Ψ r(r,μqEE y=( r →=((= → ( , , ) Ω ⋅E ∇→,tΨ + σ∂t EΨΩ − S Ψ μ η ζ Ψ , E , Ω ) ʹ ʹ ʹ ʹ ˆ q ( r , E , Ω ) = d E d Ω ( r , E → E , Ω ⋅ Ω ) Ψ (r00, E σ ˆ ee y = R → q ( r , Ω ) ˆ ˆ s E ∞ Ψ , E , Ω 0 4π r = ( x , y , z ) y,)(E ee( rsource , ee q ( r , Ω ) Ψ r , E , Ω ) ∂ ˆ yy E which is the photon resulting from → r = ( x , y , z ) ee ee →) yy eeEq., 2 ˆ (eeryyeqe, E→ee→= ,(Ω ˆ ⋅∇ → e ,ˆ → qΨeeΨe e + σ te Ψ eqe−∂→E ΨS R Ψ ee → ee → e → e ˆqyy)qγe(eer→+(,→rqE 0 4π Ω q + qee q eeqqΨ ˆ ˆ ˆ q ( r , E , Ω ) ee ˆ → r , E , Ω ˆ ˆ γ ʹ ʹ ʹ ʹ q , Ω ) ˆ ee q ( r , E , Ω ) = d E d Ω ( r , E → E , Ω ⋅ Ω ) Ψ ( r, E σ → ee → Ω ,r→E,,,Ω ,Ω Ω) )ˆˆq) ee (r ,,EE,,Ω ((r(eerree,,(E where ˆ =(r∂(,E Ψ q) s ˆ )) Ψ ,Ω )q)(q photon Ω ,ˆζq= μE,η EΨ,2γ→q interactions ˆΩ)) ,ζrζ, ,E )E,,),Ω μq,,η η(,(E ˆ→ˆΩ → ) rγe= ye( xe, y , zqΨ q r E Ω Eq. ee = ( ) μ y →ye Ω yy ˆ ee 0 4 π ˆ yy E → → r = ( x , y , z ) + q q+ q , q yee q (re,qEee, Ω) Ψ ye(qr→, E(,rΩ )→yyˆ )→→ˆ ˆ qryy=(rγγ y, E ( ,xE, y, Ω ,ˆz)) , ye E→ , σˆγγ = Macroscopic photon-to-photon differential γγ ee Ψ, Ω E ,rΩ qe (r , E σyeeeγγ , ,E ,Ω ye(ee ˆe(eyeq)r(q,→r→,Ω Ψeqyeye → → ˆ)ˆq) σ s→(→r→, E ,ˆΩ scattering ζ) ˆ)Ω ˆ) )q, (r , E , Ωs ) Ψ q yeqqqΨye σσsγγs Ψ →Ω (E,r,source, E , Ω ˆ Ψ , E ) scattering q ˆ yeE yeγye= Electron-to-electron → ( r , E , Ω ˆ q ( r , E , Ω ) q ( r , , ) ye ˆ s q ye Ψ ( r , Ω ) γγ q ˆ Ω = →( μ ,ˆη , ζ q)qye (→→(r→r, ,EE, Ω Ψyy ˆ ) γyy q q ( r , E , Ω ) ˆ , Ω ) σ y p ˆ →yy ee cross section Ω = ( , , ) y μ η ζ q ( r , E , Ω ) yy s Ψ q → ye y e γ ˆ yy q ( E , Ω ) q q ,Ω γe which isˆ )the electron ˆy )ˆq→ resulting ˆ pEˆ,ˆ q , )from y→→,σ,zeeE ) →Ω (eer ,yˆ=→E(q, xΩ γe p Ψ eyy(rq→,ysource E( E ,ˆΩ ) σ γγ qΨee y e q (qr , γE ye (,E,pzΩ ,)Ω) ) E , ppσ qE,yy,Ω ((ˆrr)rσ q),ye=Eγs(e,(→,xqΩ rˆ,p,y)y)E s ˆΩ y ,ˆΩ σσsγse (ˆ)→rˆ)),qE q ((qrE,yyE q, Ω ,y(Ψ Ω y(→r(),E Ψ , Ω p ye ˆ sγe q y qqq yyyey r→p electron e,)Ω ˆ ΨΨe qqy → interactions E → s( r , E , Ω ˆ p → q ) → q ( E , Ω ) ˆ q ( E , Ω ) q ( r , E , yy → ˆ →= q( (,ΨˆΨ E,,ζy(Ωr()r), ,ErE,→Ω ˆ ) y qqeeyy ˆ → ˆ→) ) r =ee( x, y , z ) E , σ see = Macroscopic photon-to-electron differential Ω q (r , E , Ω → ee yy ee p,Ω q (E rp μ η q ye, Ω) qqqyyeeeerp yerr→→p→ → ˆ, )Ω ˆeeσ σ ee → s,→E → ˆ )ˆ EΩ ) σσsees eeσ ye rpep → y Ωˆ =qˆ( μqye,(ηr→,(,yζrEq,),yy q r e pp → ˆ q r ( , Ω ˆ pσ sγseee production cross section q r ˆ y s ˆ ( r , E , Ω ) p q ( r E , Ω ) q ( r , E , Ω ) Ψ y = Photon-to-electron scattering source, , p q ( Ω ) E e q ( r E , Ω ) ˆ r = ( x , y , z ) Ω = ( , , ) μ η ζ q ( E , Ω ) , ) σs p , Ωe )→ q ˆr ,) EΨ Ψ q → →e ˆ) qe Ψ ( E→,e(Ω (rˆ,resulting E →, Ω ˆ ) eyy e p ˆ e ee ye which from q ee (qr , E , Ω e p ee ( μ ) p e qsource yee e ris the electron p q ( E , Ω ) → r ˆ → y→ E Ω ( , ˆ Ωʹ → ˆ yy)yeee e q p q ˆ ee e → q ye e ˆ y yy p y p Ψ ˆ q ( E , Ω ) p σ ˆ μ = μΩ q ( E , Ω ) ye e ˆ q = l Macroscopic electron-to-electron e ζ) ˆ p ˆ e p s PP μdifferential rE ,yeΩ q(,Ω Ω )Ω q q eye rp photon ˆ→)(())→rr→,, E q)ˆ)q,)→(Eqr(q,,r(Ω ,((,E ,,Ω ee 0 qqq qqqe → interactions ˆ 0⋅⋅)Ω E yy Ψ rΩ ),EE qqqqe(((E ,E,,Ω 0 = Ω ⋅ Ωʹ PPl l((μμ00)) ˆˆ ))) pp q E , Ω ˆ ˆ ˆ ( ʹ μ 00P=l (Ω q , Ω → q ( r , E , Ω q ( r , E , Ω ) E Ω ) → l ( μ0 ) eqq →p ˆ ⋅ Ωʹ ˆ r → yy r →→ → = Ω scattering cross section μ P ( ) μ p q ( E , Ω ) rrp yy → q rp r→p l 0 0 ye → ˆ The LBTE

(

(

)

) ( (((( ) )))) ( ) ( ) ( )( (

)( )( (

(

ˆ ) eeye ) q (r , E , Ω qq ey = yy → σ y ˆ qˆ (r , Et, Ω) y y σ etyeyy y Ω ) σt σ qqσtyt σ te σ te → σ ˆ t)e σ σtteeye q ee (σr , E , σ Ω q

rp

(( ( (

)) ) )

(

)

)

)

) )

∫∫

∫∫

∫∫

∫∫

∫∫

∫∫

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫



) t σ ty (r , E σ e → σ t (r , ES)R 10



t

(r , E )

S Rt e σσ t t

SσRt SR → S (r , E )

y

∫∫

∫∫

p

qˆ (ˆr , E , Ω) p p ˆˆq)→yyeyee( qE→→e, (Ω p q e( E , Ω y er ppp ˆ ⋅ Ωʹ E,))ye,Ω Ω ,ˆΩ ˆ)→))→point source pp p q (σ E ,yΩ μ0 = Ω Pl ( μ 0 ) qrq), (E(E(r)r, Ω Extraneous photon source, for ,EˆE qqqqyeeey q →→ ee ˆΩ ( ˆ) )→ → → → → q ( r(,rE ,Ω t y The q E , , basic assumptions used in Equations 1 and 2 are briefly →y → y y y ee yr p pq, at ( r , E ) (r→→,→),Ω E))→ σ yt ˆ(→r ,pE ) σ t p → σ erp (ert(yy,E rrpp σrpt qye ( E , ˆΩ yE ˆe ()σ σqposition tq rrqr→,,ˆee,E ))r) ,σEt, Ω σ yt p, E ) ( ) ( r ( E σ t t( summarized as follows: Both charged pair production secr , E ) σ → q ( E , Ω ) → σ → ( E σ t y e p y e ye e ˆt → ˆ → ye t (→ E σ t (r , E ) r , E) r→→→)), →E ) σˆ ete (→r→, E ) σ(Er)te,,e(Ω (r,,Ω qq yσ ye σr e → e qq (σ t er→ E ondary particles are assumed to be electrons instead of one tσ → ( r , E ) q ( r , E , Ω ) et rp e p → σpt ( r , E y → ( r , E ) σ ) Thiseyeσσsource represents all photons coming from the ˆ e ( r , E ) σ p yt ˆt( r , E ) → t E→, Ω→→q) e (σ →→ ˆσ(r))ye,(→E t tΩ ) )σ, Ω σ tˆ ( rp qe σ tttey→(pr→, E ) rp σ t qyσ ((σ σ, Et ()r , E ) p electron and one positron. Also, the partial coupling tech(Er→q→), E()EEσ,t,Ω tyt t σ →,EE q r , q ( r ) t ˆ r , σ t machine source model. t → σ q ( tEσ,σΩ ) E)) ˆ σeSt (R σ t σqσt → py σ ter , )E ) e → ) p → → →r(,rE → EE), Ω σ tt ((tq(rrry,, (,E t et ˆ ) →S R q ) σtty→p ,Ω σy ) (→r , E ) σSσ(tyr→Rt(,(r(→Err→→,p→),,EE is assumed, rrp S R (r , E ) whereby photons can produce electrons, y (r , E E) ) SpR (nique e ˆS) σ → r , E ) σ ˆ SσqRtσ y σ t ˆt( r , E )S RS →p S R q ye (r , E , Ω te t → r σ S t y q E Ω ( , ) ( r , E ) e → R p q→ (( E E,, Ω Ω , E )electrons do not produce photons. Regarding the latter, ˆ )) ,→(for (r→r, ,EE)) SpR (rbut R SSRt = Extraneous rp → source, electron r , Epoint ) SSReR(source qqe SσRte→ e → SσReR ) σyσ σ t (qr , E → (rtr,,EE)) yy σ→→Rt (r , E ) →S R ( r , E ) σ ty r t p σ t ( σ → y ˆ r p ) , at positionrp p the from Bremsstrahlung photons is assumed to be t σtt Ω) → σ t (r , E → → r , Eenergy ) σσttyσ S σe t →( r ,σEt )(eSr →,→RE()rp, E ) SpR (negligible q e (E (ˆrq),eE( E) ,σΩ σ,t Ω qS eRt e ˆ e e Rt y → and is discarded. σ ( r , E ) e ) ( r , E ) σ qσ et → σ t (r , E )σ t → (r , E→the ) σ ttfrom yt → σ te Ssource represents all electrons → rp S Rp (r , E ) → e σ coming ( r , E ) ˆ y → S ( r , E ) → t σRty This q ( E , Ω ) y R e → S → rp σ t ( r , E )σσ t((rr,,→EEσ)) t (r , E ) σσqt t R source σ t (r , Eσ) t machine These assumptions have only a minor effect on the energy → model. σp t σt te →(r , E ) e (Sr→, E()r→, E ) →→ ˆ →) rp σS te ,Ω e σ y deposition field, and are similar to those employed in clinical S ( r , E)) R t σ ˆ σ t (r , E ) y → S RR(r , E σ S→St RyR σσ tR = Macroscopic ˆ ʹʹ ) t →(r , Eσ) ty (r→, E ) , photon totalσcross section, t ( r , Eσ σ Monte Carlo codes. A primary assumption of Equation 2 is ( r , E ) t → t → t e S (σrt, E ) -1 y σ σσ)tye ((Srr→,R, EE(r)→) , E ) → that the Fokker-Planck operator (of which the CSD operat (r , E σS te R units e of cm σ t t σ R S (r , E ) → r

∫∫

R e → σ t (r , E ) → σ t ( r , E ) σ t (r , E→) → σ t ( r , E ) S R (r , ES) (r→, E ) R → S R (r , E )

ˆʹ ˆʹ ˆʹ



σ see

φl , m ( r , E ʹ)

ll ≤≤ 77

→ * ∞ ∞ ˆ ˆ →→ →d →Ω' Yl , m (Ω ʹ) Ψ ( r , Ω' ,E ), γγ γγ→ → γ γ∫ ˆ, Ω ˆ, Ω ˆ⋅ Ω ˆ, Ω ˆ ˆ ˆ ʹ ∫ʹ d∫Ω ʹ σʹ σ ʹ ʹ ʹ ʹ q γγq γγ(r(,rE, ,EΩ )ˆ =) =∫ d∫EdE ( r , E → E , Ω ⋅ Ω ) Ψ ( r , E , Ω ) , ʹ ʹ ʹ ʹ dΩ ( r , E → E ) Ψ ( r , E ) , s s 4π Eq.88 ee Eq. σs 0 0 4π 4π

γ γ ˆ, Ω ˆ, Ω ˆ⋅ Ω ˆ⋅ Ω ʹ ∫ʹ d∫Ω ʹ γe γe ʹ ʹ →E ,EΩ ʹˆ)Ψ ʹ, Ω ʹˆ)ʹ ), , q γqe (γer(,rE, ,EΩ )ˆ =) =∫ d∫EdE (r(,rE, E ʹ)Ψ ʹˆ, Ω σ deeΩσʹ σs s(r(,rE, E→ →→

→→

0 0

4π 4π

→→

s

Eq. 8

tor is the first ∞ ∞ order term), is used for Coulomb, or “soft”,

ee e Catastrophic ˆ, Ω ˆ, Ω ˆ⋅ Ω ˆ⋅losses. that ineesmall ʹ ∫ʹ d∫result ʹ σʹ σ ʹ→ ʹˆ)Ψ ʹ, Ω ʹˆ)ʹ ), , q eeq (eer(interactions ,rE, ,EΩ )ˆ =) =∫ d∫E Ω Ω (er(,rE, E ʹenergy ʹ)Ψ ʹˆ, Ω dE dΩ ,E →E ,EΩ s s( r(,rE →→

→→

→→

Eq. 8 interactions in large energy losses are represented 0 0 that 4π 4result π with the standard Boltzmann scattering. This can be seen as the deterministic equivalent to electron condensed history Eq.Carlo. 8 models in Monte →

ll == 00

l=0 Clinical Perspectives | Acuros XB l ≤ 7 ˆˆ

/ ee →→ γγγγ/ γ/ eγe/ ee

qq

Eq. Eq. 88

∞ ∞

ll ≤≤77



((rr, ,EE, ,ΩΩ)) 77

l l ∞∞

→ →→ ˆˆ )) / ee →→ qqγγγγ/ /γγee/ /eeee((rr,,=E=E,Σ Ω ˆΩ ˆ) ,Σ Ω dE dE'σ 'σsγγ,sγγl,/l γ/ eγe/ ee ((rr, ,EE' → ' →EE)φ )φl ,lm,m((rrl, ,E=E')0')YYl ,lm,m((Ω Σ Σ ∫ ∫ l≤7 ∞ m ==−−l 0l 0 77 l =l = l0l0m∞ → → → → ˆˆ).). == Σ dE dE''σσsγγsγγ,l,l/ /γγee/ /eeee((rr,,EE''→ → EE))φφl l,m,m((rr,,EE')')YYl l,m,m((Ω Ω γγ / γe / ee → Σ Σ Σ ∫ ˆ) ∫ q ( r , E , Ω Step 1: Transport of source model fluence into the patient l l==00 m m==−−l l00 l=0 7 l ∞ yy → and qqee ˆ l ≤ 7= photon / γe / ee → sources,qq and The external and dE 'σ sγγ,electron ( r , E ' → E )φl ,m ( r , E ')Y,l ,are Σ Σ l m (Ω). ∫ → modeled aslγγ=anisotropic sources in Acuros XB. yy ee At each ˆpoint q 0/ γme =/ ee− l (0 r , E , Ω ) qq and and qq →

static beam phase space (i.e., control point), a separate point 7 l ∞ → / γe / ee → For the primary source exists for each of thedE AAA ˆ ). =ΣΣ 'σ sγγ,lsources. ( r , E ' → E )φl ,m ( r , E ')Yl ,m (Ω y e q and q source, the anisotropy of is described through a 2D flu0 l = m = − l γγ / γe / ee 0 ˆ) q (r , E , Ω ence grid, in which both the particle fluence and energy To represent the anisotropic behavior of the 7differential l ∞ → qy γγ / γe / ee → spectra are spatially For the extra-focal and wedge ˆ = Σ Σ dE 'σ s ,l ( r , E ' → E )φl ,m ( r , E ')Yl ,variable. scattering and production sources, in a mathematically m (Ω). y e q scatter sources, the anisotropy of q isand described through a l =0 m = − l 0 practical manner, the macroscopic differential scattering qy 3D fluence grid, and the energy spectra is spatially constant. qqee cross sections are expanded into Legendre polynomials, For the electron contamination source, the anisotropy of ˆΩ ˆ⋅ Ω⋅ Ω ʹ. ʹThis expansion allows the PlP(lμ(0μ)0 ), where μ 0μ 0= =Ω qqeeenergy q y and q e is described through a 3D fluence grid, and the differential scattering or production cross section(s) to spectra is spatially constant. All point sources are located at be expressed as: →→ ye ˆΩ) ˆ the target for the respective control point. rrpp qqγγ((EEq,q,Ω)





Eq. 6 γγ ˆˆy located at position, r→r→ , Ω) For a photon point source,qq ((EE,,qΩ) Eq. 6 pp / γe / ee →ˆ ʹˆ ˆ ˆ eeσ→γγ γγ / γe /Eq. → γγ ⋅/Ω γeʹ/) ee ( r , E ʹ → E , Ω ʹ) Eσ, Ω ˆ ⋅Ω ˆ ʹ) Equation 1 becomes: (r6, E ʹ → E(,rΩ, E⋅ Ω→ σs e Eq. q Eq. s66 s → γγ → ˆˆ→⋅∞Ω ˆˆ ʹ) ee γγ ∞// γγee // ee 6 ˆ ∞ 2l +σ E ʹʹʹ/→ ((rr→→,,→γγ ,Ω +ee1 rp q γ ( E , Ω) ˆ ,⋅⋅E ˆʹ2ʹʹ)→ e /E ee γ→ E → Ω σσ 1=sssγγγγ/ γ/ e2γel/ ee / r(r, E E EE,,(Ω ( Ω Ω → Eq. 9 l) +) 1E ) P r σ γγl (/μ γe0/)ee → , Σ ʹ γγ / γe=/ eeΣ( r , E ʹ → E P , → ) ( σ μ Eq. 9 s,l ˆ ˆ ʹ =Σ (r , E → E ) Pl ( μ 0 ) l 0 σ s,l E , Ω4⋅πΩʹ) ∞ σs s,l ∞ → + Eq. ʹ9 → γ // γγee // ee ∞ 2lll π ++111σ γγ l = 0 4π l = 0 γ) γ γγ γ ee γγ = 0 224 e Ψ ˆʹq⋅→ ˆ )δ ( →r − →r ) , = ( rr→→,, E E )) P / e / ee γγ γ Ω ∇ +ll (((σμ == lΣ ( E → E P )) = q + q ˆ(E,→Ω σ μμt000Ψ ∞ 2l + 1 s,l p Σ ʹ ( r , E → E ) P σ s,l → ˆ )δ ( →r − →r ) , 4 π l γγ / γe /l ee s,l Eq. 9 Ω ⋅ ∇Ψ γ + σ tγ Ψ γ = q γγ →+ q γ (E, Ω γ ( E , Ω) = 00( r44,π p ˆ π ʹ = E → E ) P ( ) σ μ = l Eq. 9 → r q Σ l 0 = l 0 → / ee angular where s,l in the scattering → p → → ˆ ⋅Ω ˆappearing Similarly, γ γ ʹ γ γγ γ E, Ω (r , E ʹ →fluence σ sγγ / γethe ˆ ˆ → Eq. 9 → ( r l = 0) 4π − Ψ + σ t Ψ = q + q (E, Ω)δ rp ) , ˆ ⋅ ∇Ψ γ +where ˆ )δ ( r − rΩ)⋅,∇ Ω Ψ γ = q γγ + q γ (E, Ω σ tγwhere → → source is expanded into spherical moments: pˆ → γ γ γ γγ γ ∞ 2l + 1harmonics ˆ Eq. 9 Ω ⋅ ∇Ψ + σ t Ψ = q + q (E, Ω)δ ( r − rp ) , γγδ9/ γe / ee → ʹ Eq. → where =Σ (rγ, E →ˆ E ) Pl ( μ 0 )where σ s,l → → → δ → = Dirac-delta function γ γ γ γγ γ ˆ ˆ Eq. 7 ˆ r q ( E , Ω) ˆ 4 π ∞ l Ω ⋅ ∇Ψ γ where ʹ γ γ γγ γ Eq. 7 ʹ p ˆ ˆ )δ (+→rσ−t→r Ψ) , = q + q (E, Ω)δ ( r − rp ) , l =0 → ∞ → l Ω ⋅ ∇ Ψ + Ψ = q + q (E, Ω σ ∞ l ˆ ˆ p t → → δˆ ʹ)Y m (Ωʹ) , Eq. φl ,m→( rˆ, E → ˆ Ψ) (=r , E ʹ, Ωʹφ) = (δr , E ʹ)Ψ Eq. Ψ777( r , E ʹ, Ωʹ ,ˆ )l ,= where Eq. E ʹ), Ωʹ Ωʹ) ,principle of linear superposition may be used to define l ,m l = 0 m = − lYl(,mr(,Ωʹ ∞ l φl ,m ( r , E ʹ)Yl ,m (The where ∞ ll → → 7 ∞ l =0 m = − l → → ˆˆ ) = l =0 m =− lφ ( r→, E ʹ)Y (δthe ˆˆ ) photon →, E ʹ, Ωʹ Ψ ( r Ωʹ angular fluence as the summation of uncollided ˆ→ )) == ˆ )),,, φφll ,,mm ((rr,, EEʹʹ))YYlll,,,mmm ((Ωʹ ∞Ψ Ψ((rlr,, E Eʹʹ,,Ωʹ Ωʹ Ωʹ → where δ l = 0 m = −ˆl l ,m Eq. 10 ˆ 0 l = m = − l and collided fluence components, ʹ ʹ Ψ( r , E , Ωʹ) = φδl ,m ( r , E )Yl =l0,mm(=Ωʹ −l ) , where Eq. 10 γ γ where ˆ) where . Ψ γ ≡ Ψunc + Ψcoll l =0 m = − l Ω where γ γ ˆ) . Ψ γ ≡ Ψunc + Ψcoll Yl , m (Ω Eq. 10 ere Eq. 10 ∞ l functions = Spherical harmonic Eq. 10 → → ˆ γ γ γ ˆ ˆ Y ( Ω ) ˆ ʹ ʹ Ψ ( r , E , Ωʹ ) = ( r , E ) Y ( Ωʹ ) , φ Ψ ≡ Ψunc + Ψcoll . where γ γ γ ,, m Eq.Ψ10 ˆ )) Y (Ω → l ,m l ,m lY,lllm Ω ≡ Ψunc + Ψcoll ,m m (→ ,. , E(ʹ)Ω where l =0 m = − l γ γ ˆ) = Angular indices . Ψ γ ≡ Ψunc + Ψcoll ,m Eq. 10 φlll,,,lm m ,m m ( r→, E ʹ) e Eq. 10γ → where γ γ γ → * Ψ ˆ ˆ ʹ m ) φφl , m ((rr→,, *E . Ψ ≡ Ψ + Ψ where ʹ γ γ γ unc ' Y→l ,m (ˆΩ )Ψ ( r , Ω' ,E ), γ unc coll where → Ψ ≡ Ψunc + Ψcoll . φlld,,mmΩ('rY, EEʹ(ʹ))Ω ˆ dʹ= ΩSpherical harmonics angular Ψunc → ˆ ʹ)Ψof ˆ ' ,E where dΩ'moments Yl*,m (Ω ( r ,the Ω ), l , m 4 π ) Ψ ( r , Ω' ,E ), ˆr), E ʹ) ( (Ω , m → γ m calculated as: 4fluence, π ˆˆ ʹ)Ψ ( r→→, Ω ˆˆ ' ,E ), Ψunc = Uncollided, or unscattered, photon angular fluence. γ 4π d ' Y ***m ((γΩ where ˆ ʹʹ)Ψ ( r , Ω ˆ ' ,E ), ddΩ Ω Ω Ψunc Ω''YYlll,,Ψ m (Ω ) Ψ ( r , Ω' ,E ), ,m → where m γγ * 4 π ˆ ˆ coll Ψ ʹ 4 π d Ω ' Y ( Ω ) Ψ ( r , Ω ' ,E ), Refers to photons which have not yet interacted 4π Ψunc l ,m → coll ʹ ( r , E ) γ 4π m γ Ψ with the patient/phantom. γ unc γ Ψ Ψunc coll Ψcoll ee * denotes the *complex → where conjugate γ σs ˆ ʹ)Ψ ( r , Ω ˆ ' ,E ), Ψcoll = Collided, or scattered, photon angular fluence. RedΩ' Yl ,m (Ω ee σ γ ee 4π σσ ssee l = 0 Ψ γ = Macroscopic electron-to-electron s fers to photons which were produced or scattered ee coll Ψ7colll = 0 differential l ≤ l = 0by a photon interaction in the patient/phantom. s scattering cross section Eq. 11 l≤7 l≤7 = 00 Eq. 11 →lll = ˆ ⋅ ∇Ψ=γ 0 + σ γ Ψ γ = q γ (E, Ω ˆ )→δ ( →r − →r ) , Ω l ≤ 7 unc Equation t unc 10 into Equation e Equations 6 and 7 are exact. Additionally, for purely isotropic γ 9,p+leads γ the γ (E, Ω ll l≤≤=770 Eq. Substituting to ˆ ˆ )δ ( →r − →r ) , Ω ⋅ ∇Ψunc = qfollow11 σ tγ Ψunc p Eq. 11 l ≤ 7 → scattering, l = 0 is also exact. However, Acuros XB sets a → → → ing equation for the uncollided photon fluence: γ γ γ γ ˆ ˆ → Eq. 11 → ( r − rp ) , ⋅ ∇Ψunc + σ t Ψunc = q (E, Ω)δ γ + γ Ψ γ γ = q γ (E, Ω ˆ of ˆ )δ ( r − r ) Ω → the scattering order, l ≤7, and hence the number / γe / ee on γγ limit Ω ⋅ ∇Ψunc σt Ψ ˆ) → → p ,ˆ → γγ unc 8 (r , E qEq. ,Ω γ γ γ unc ˆ l = 0 Eq. Eq. 11 11 Ω ⋅ ∇Ψ Ψunc → unc + σ t Ψunc = q (E, Ω )δ ( r − rp ) , ˆ8) / γe / ee → harmonic in the 11scattering/producˆ ) Eq. → q γγ / γe / ee ( r , Eq. E ,spherical Ω 7 l ∞ q γγmoments ( r →, Ekept ,Ω Eq. l ≤ 7→ γ γ →+ γ Ψ γ = q γ (E, Ω → Eq. 88 ˆ ˆ )δ ( →r − →r ) , → / / e ee γγ γ → Ω ⋅ ∇ Ψ γ ˆ / γeLegendre / ee → γγ Using addition theorem, the scat- γ p ˆ ⋅ ∇Ψ γ + σ Ψ γ = q γ (E, Ω ˆ )δ Ψ unc = dE q' σthe (→r, 7E , E, Ω 'ˆˆl→ )σ, t unc γγ 7 tion l ∞ source. ( r unc − r Ω ∞ E )φ l , m ( r , E ')Yl , m (Ω). 8 r ) ee ( γγs ,//lγγee // ee Ψ p unc unc t ˆ q ( r , E , Ω ) → → unc γγ=/−γle / ee q → ( r)φ, E ,become: Ωr ), EEq. γ 0 m /ˆγe). / ee → γγ(Ω 12 →production 0 ˆ tering sources = ( r , E ' → E ( ') Y σ / γeand /l '= ee γγ dE Ψ ˆ = dE ' ( r , E ' → E ) ( r , E ') Y ( Ω ). σ φ , , , s l l m l m ∞ unc 7 l ∞ s ,l l ,m l ,m q (r , E , Ω) 7 l ∞ →Eq. 12 γ l =0 m = − l 0 / e / ee → ˆ γ) e −τ ( r , rp ) ˆˆ ).ˆ of Equation l =70 m =l− l 0 dE ' σ γγ γγ //γγγee//ee qthat Ω ( E ,Ψ →(Ω = rr→→,, E '' → E ))φφl ,m ((rr→→,, E ')')γY ee ( −τ ( r , rp ) A property 11 is can be ,solved for anaˆ ˆ γ ,l sγγ l(, m γ ˆ = dE ' ( E → E E Y ( Ω ). σ 7 l ∞ ˆ unc Ψ r E Ω = Ω − Ω , , ) δ = dE 'σ ss,,ll ( r , E ' → E )φll,,mm ( rEq. , E unc ')12 Yll,,mmΨ(unc Ω). → r , rp Eq. 8 → γ =→ −l 0 γγ / γlle==/00eem → → 2 ˆ ˆ ) =expression ˆ for qthe( E , Ω) e ˆ ). 4following π(r , E , Ω , Eq. Ψ Ω−Ω δ = dE ' ' → E ) ( r , E ') Y ( Ω σ φ y s ,l 12 lytically. Doing so provides the l =0 m m(e==r−−,ll0E unc r r , r − r 0 , , l m l m → p → γ p ˆ ) er −−τ ( rr→, rp )2 ˆm =)− l 0 q and q 4Eπ, Ω −τ ( r , rγp ) → Eq. 12 q ( γ q γγ / γe / ee ( r , El =,0yΩ ˆ ˆ ˆ ˆ p q ( Eangular , Ω) e fluence from Ψunc ( r, , E Ω r , rp γ , Ω) a=point δ Ω −source: ˆ ) =uncollided ˆ −Ω ˆ photon where q y and q eΨ γ ( r→, E , Ω q l and qe ˆ ) e→−τ ( r→, rp )2 , δΩ ∞ unc r , rp q (4Eπ, Ω 7 Eq. → →γ 2 → where r − rp , y 12 ˆ ˆ ˆ y → e 4 π Eq. 12 γγ / γe / ee → Ψ r E Ω = Ω − Ω ( , , ) δ ˆ ). q r − runc r , rp γ = dE 'σ s ,l ( r , E ' → E )φqlqq,myy( rand , E ')qYe (Ω p ˆ ) e −r→τ (−r , rr→p ) 2 and −τ ( r , rp ) and qq el ,m y q ( E4,πΩ →ˆ γγ ˆ ˆ ˆ where y y e p , l =0 m = − l 0 q unc( E( r, Ω e ) = δ Ω − Ω r ,r q Ψ , E) , Ω γ →→ → ˆ q and q where ˆ −q Ω ˆ p → → 2 , r ,rE , Ω) = δ Ω yΨuncr( − r , rp 4 π 2 ˆ y → → r − r → → → → where Ω r , rp q yy p ˆ → →p r pqqqand r 4rπ− rp r − rp → → Ω y r , rp y y r r q r − r where → and → p q y e where → → ˆ q p , r − rp → → Ω 11 ˆq and q r , rp → r − rp → qq yyy Ω → → , r p and r r , rp → → q r r ˆ r − r and → → → → p Ω r − rpp → → qr , rypy r − rp r p and r → → τ (r , r p ) ˆτ (→r→q, r→→ ) r r→−−rpr→p , , Ω qe → →

ΣΣ

ΣΣ

ΣΣ Σ ΣΣ Σ

ΣΣ

ΣΣ





∫ ∫∫





ΣΣ ∫

ΣΣ ∫

ΣΣ ∫

ΣΣ ∫ Σ ΣΣ Σ ∫∫

(

(

ΣΣ ∫





→ →

→ →

→ →

→ →

→ →



→ →

)

→ →

)

(

→ →







)

→ →

(

( ( (

















)

) ) )

→ →

→ →

→ →

→ →

q

Eq. 11 Eq. 11

ˆ ⋅ ∇Ψ γ + σ γ Ψ γ = q γ (E, Ω ˆ )δ ( r − r ) , ˆ ⋅ ∇Ψ γ + σ γ Ψ γ = q→γ (E, Ω ˆ )δ ( →r − →r ) , yyΩ Ω p unc → t unc → q coll unc t unc ˆ γ →pγ γ + γ (E, Ω → ˆ → ) ( r − γ r , Ω ⋅ ∇ Ψ Ψ = q ) σ δ γ γ γ → unc ˆ ⋅ ∇Ψ + σ Ψ ˆ )δ ( rt γ− runc p → → ), Ω Ω γ γ γ p ˆ q⋅ ∇(E, ˆ unc t unc =Ω ) Ψunc + σ t Ψunc = q (E, Ω)δ ( r − rp , γ → Ψγ Ψ γ γ γ γ unc ye unc ye ˆ ⋅ ∇Ψ + σ γ Ψ ˆ )δΨ( →rγ − →r ) , Ω = q (E, Ω q q coll coll t unc unc Eq. unc 11 Ψunc γ p Ψunc→ ye ye → → qunc γ→→ →+→ γ Ψ γ = q γ (E, Ω q coll ˆΩ ˆ )δ ( →r − →r ) , γγ ++ γγΨ γγ ==qqγγ (E, ˆΩ ˆ ⋅⋅∇∇→Ψ ˆ∇ ˆ))δΨ Ω ⋅ → ) ) →σ ( ( r r − − r r , , Ω Ψ Ψ (E, Ω σ σ δ p unc unc γ t γ γ γ pp unc + σtt Ψ unc unc Eq. 12 ˆ ⋅ ∇Ψunc ˆ Ψγ Ω unc t unc = q (E, Ω )δ ( r − rp ) , Eq. 12 unc −τ ( r , rp ) ˆ ) e −τ ( r , rp ) γ e ˆ q γ (E, Ω q E Ω e ( , ) → Ψ e ˆ Ψ −γ τ ((r ,r→rp,)E , Ω ˆ) =δΨ ˆ −Ω ˆ Eq. 12 Ψ γ ( r , Eγ,γΩ γ ˆ)=δ Ω ˆ −Ω ˆ → γ , Ω γ , unc r r , r r − ( , ) τ q E Ω e ( , ) p p p unc → → 2 ˆ )unc γ r , rp q ( E Ψ →ˆ →ˆ 2 ˆπ), Ω Ψunc −τ ( r , rp ) , 4π e → Ψ γˆ γ 4 e Ψ r E Ω = Ω − Ω ( , , δ γ unc ˆ r − r ˆ ˆ unc r r , r − r p q 4(π E , Ω) →e → 2 Ψ Ψunc ( r , EΨ, Ω ) = δ Ω − Ω rγ ,rpp → pˆ ,p unc ˆ ˆ − 2 Eq. Eq. 12 → Ω Ψunc ( r , E ,4Ω δ→Ω r→14 −14r→p 2 , Eq. Eq. πτ ()r ,=r 14 r − r p r , rp where 4 − ) π γ p r −→ rp e ˆ) e ∂ ˆ →e e ee e γee ∂γe q (E, Ω → γe γe γ where ˆ 14 ˆ)=δ Ω ˆ −Ω ˆ SΩ q ee , = q ee + q coll Ω ⋅ ∇Ψ + σ te Ψ e − = q + σ+t qΨcoll −+ q uncS+R Ψ , Ψunc ( rEq. , E ,12 Ω ⋅ ∇Ψ + q unc + qe Eq. RΨ r , rp → → 2 where ∂ E E ∂ 4π γγr − r → r r − ( , ) τ γ p −−ττ( (rr, r, rpp) ) ˆ )→e pˆ ˆΩ qwhere ( E→, Ω ˆ ⋅ ∇Ψ e + σ e Ψ e − ∂ S Ψ e = q ee + q γe + q γe + q e )ˆ γ γ((E Er→,,,Ω → Ω ˆeee) −=τ ( rδ, rpΩ ˆ γ→ γ →→ ˆ),))Ω t R coll unc ˆˆ)) ==δδ Ω ˆΩ ˆ −−Ω ˆˆ→ →→Ψqqqunc ˆ (where r − rp→ → 2 , → Er ,EΩ ,,− Ω Ψunc Ω Ω ((rr→,p,EE,,Ω → r , rp where Ψ where r γ− Ω → ∂E unc r r r r , , ˆ ˆ ˆ r , 2 2 p p → → → → , Ψ→unc (→r , E , Ω) = δrΩ and −Ω 4π→ →r − r r p and r rr , rr 44ππp ˆ r − p p → →− → 2→ → → r r − r r Ω p p → → 4π ye rr −and where yer − r rppp r→ ˆ r , rp = rr −−rrpp , where r→ and r r→−→are rp the qsource → qunc p and Ω unc → desti= p ,First scattered electron source. Refers to electrons r , rp where → → p and , r − r r r p and → → p r − rp → → , ye − rp respectively. of the rayr trace, which are created or scattered from the first pho,→ qunc r − rp nation points → , → → ye r r ye ( r , r ) τ and ton interaction inside the patient/phantom. → →→ p p → q coll → →q coll → → → → → → )ˆp= rThe τ (rr→, −r→prΩ optical (measured in mean-free→ distance → r − r →− → → → r − r r , p r and r p . → → → → pp , rpr − r r and rrr→p . ye τ (r , r p ) r paths) q coll rr→. → r→→and r→ . r p and r = Secondary scattered electrons source. Refers to → →pp and and → → → → pbetween p → rr→−−rrp→p r and r pr p. and r r −r→rand p electrons which are created or scattered from , rp . r − rp ,,

1

( (

(

2





(((

→ →



→ →

) )



→ → → →

→ →

)

→→ →→ →



)))

( (

→ → → →







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)

→ →

→ →

→ →

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→ →

γ Ψunc p))) γ p

Ψunc

γγ

γγ qunc Ψ Ψunc γ Ψunc unc

Ψe

, for each primary, extra focal, and Equation 12→is solved → γ → → r p .calculation, to compute Ψunc r and wedgeτsource in the through(r γ, r p ) Ψ unc out the patient. The electron contaminant source is modeled → → γγ → → → → γγ Ψγ r and r p . qunc → and and rrp→p..the inclusion of the unc γγ inqγγ a similarunc manner,rrrbut with CSD operaand rp . q unc quncto account for charged γγ particle γ tor interactions. qunc Ψcoll γ γ Ψcoll Ψ γ γ coll Ψcoll Ψunc Step 2: Transport of scattered photon fluence in the patient γ γγ Ψ γγ γγ q coll qqunc Once unc is calculated according to γγ Equation 12 is solved, qunc unc γ Equation 8, and is consideredγγa fixed source in Equation 13, Ψcoll Ψ Ψcoll γ which is solved to compute Ψcoll coll throughout the patient:

secondary photon interactions inside the patient/ phantom.

Discretization methods γ

Ψcoll

Acuros XB discretizes in space, angle, and energy to iteratively solve Equations 12 through 14, the methods of which are discussed below. Spatial discretization

The computational grid in Acuros XB consists of spatially variable Cartesian elements, where the local element size → ˆ ⋅ ∇Ψ γ + σ γ Ψ γ = q γγ + q γγis adapted γ γ γγ γγ to achieve a higher spatial resolution inside the Ω unc , t coll coll coll + σ t Ψcoll = qcoll + qunc →, coll γ γ γ γγ γγbeam fields, with reduced resolution in lower dose, lower ˆ Ω ⋅ ∇Ψcoll + σ t Ψcoll = qcoll + qunc , gradient regions outside the beam penumbra. Commonly where Eq. 13 where → γ 13 γ γ γγ Eq. γγ referred to as adaptive mesh refinement (AMR), the mesh where ˆ ⋅ ∇Ψ Ω + σ t Ψcoll = qcoll + qunc , →yy → coll γ γ γ γγ γγ q is limited toγrefinement γ in factors γγ of 2γγ(from one level to ˆ ˆ ⋅ ∇Ψ Refers to photons Ω ⋅ ∇unc Ψcoll= +First σ t Ψscattered = qcollphoton + quncsource. , Ω + σ tγ Ψcoll = qcoll + qunc , coll coll yy the next) in any direction, allowing for localized refinement q which are created or scattered from the first phowhereunc to resolve areas of sharp gradients. Spatial discretization yy ton interaction inside the patient/phantom. where q coll is performed through using a linear discontinuous Galerkin yy yy qunc q coll = Secondary scattered photon qsource. Refers to phoyy finite-element method [ref. 1], providing a linear solution unc tons which are created or scattered from secondary variation throughout each element, with discontinuities yy photon interactions inside the patient/phantom. permitted across element faces. The first scattered photon q coll ye yy ye q coll q coll and first produced electron sources, obtained from solving q coll ye ye ye q Equation 12,qare also represented as linear varying functions Step 3: Transport of scattered coll electron fluence unc quncin ye in each element, the patient qunc since these sources are used for the linear e ye discontinuous discretization of Equations 13 and 14. To acOnce Equation 13 Ψ is solved, q coll is calculated according to ye ye e curately integrate these first scattered sources, the analytic q coll q coll ye Equation 8, and isΨconsidered a fixed source in Equation 14. qunc ye solution is computed at a density inside the primaryyebeam Eq. 14 qunc Similarly, from the solution to Equation 12, qunc is calculated → and penumbras of at least 8 ray traces per output grid voxel. ∂ → e e e e ee e e e γ γ e ∂ Eq. 14 ˆ e ee e γe 8,Ω and isγealso ⋅+∇qΨ ++σqconsidered SaR fixed Ψ = q + q coll + q unc + q , ˆ ⋅ ∇Ψ eaccording t Ψ − Ω + σ te Ψ e −to Equation SΨ RΨ = q unc + q ∂,E → coll e ∂ compute e14 is solved e e γe γe ∂E source in Equation 14. Equation to ˆ Ω ⋅ ∇Ψ + σ t Ψ − S R Ψ e = qΨee + q coll + q unc + qe , where E ∂ the patient: Eq.throughout 14 where → ye e e Eq. e 14∂ e ee e γe γe ˆ Eq. Eq.1313 → Eq. ˆ 13 Ω ⋅ ∇Ψ γ

qunc Ω ⋅ ∇Ψ + σ Ψ − S Ψ = q + q coll + q→unc + q , → ˆ ⋅ ∇Ψ e + σ eyeΨ e − ∂ S Ψ e = q ee + q γe + q γet + q e ,∂E R ˆ ⋅ ∇Ψ e + σ e Ψ e − ∂ S Ψ e = q ee + q γe + q γe + q e , Ω Ω t R coll unc t R coll unc q unc E ∂ where ∂E ye where q coll ye ye qunc q coll ye qunc ye q coll

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ye q coll

Clinical Perspectives | Acuros XB

Energy discretization

Step 4: Dose calculation

Energy discretization is performed through the standard multigroup method [ref. 1], which is used in both the energy dependence of Equations 12 and 13 and the Boltzmann scattering in Equation 14. In energy, the energy derivative of the Eq. 15 in Equation 14 is discontinuous slowing down (CSD) operator cretized using the linear discontinuous finite-element method Eq. Eq. 15 15 [ref. 3]. The Acuros XB cross section library includes 25 photon energy groups and 49 electron energy groups, although not all groups are used for energies where lower than 20 MV.

Once Acuros XB solves for the electron angular fluence for all energy groups, the dose in any output grid voxel, i, of the problem is obtained through the following: Eq. 15

Eq. 15



∞ e → ˆ σ ED (r→, E ) Ψ e (r→, E , Ω ˆ) . e D dE d = Ω ( r , E ) σ → i ˆ )∞∫,. ∫ ˆ ED Eq. Eq.15 15 Di = ∫ dE ∫ dΩ Ψ e (r , E , Ω →) → eρ (r 0 4π ρ (r∞∞) e →→ ˆ σ ED (r , E ) ∞e∞(r→, E , Ω ˆ )σσ. eED 0 4π → dE eei =→ ˆ ED ((rr→,,EE))Ψ ∫ E)) ∫ deeΩ→→ ρˆˆ(r→)DDi =Ψ σσDED ˆΩ dE d = Ψ ED((rr,,E dE d Ω ˆ ˆ → D Dii == ∫∫dE dE ∫∫ddΩ Ω Ψ E,,Ω Ω)) .. i ∫∫ ∫∫ 4Ψ π ((rr,,E where → →0 where ρρ((rr)) 00 44ππ ρρ((rr)) 00 44ππ

Eq. 15 ∞

where e

e where where σ ED Angular discretization ρ ee σσED For the spatial transport of the scattered particle field, the ED ρρ discrete ordinates method is used to discretize in angle e [ref. 1]. The discrete ordinates method consists of requiring σ ED Equations 13 and 14 to hold for a fixed number of directions, σ e ˆ . These discrete directions are chosen from an angular ED Ω n quadrature set that also serves to compute the angular integrals in Equation 5 for the generation of the scattering source. Square-Tchebyshev legendre quadrature sets are used and the quadrature order ranges from N=4 (32 discrete angles) to N=16 (512 discrete angles). The angular quadrature order varies both by particle type and energy. Higher energy particles have longer mean free paths, or ranges for electrons, and thus for each particle type, the angular quadrature order is increased with the particle energy.

σ ED = Macroscopic electron energy deposition cross secwhere where tions in units of MeV/cm ρe σ ED ee σσED ED ρ = Material density in g/cm3 ρρ e σ ED ρ Acuros XB supports two dose reporting options: dose-toρ e eσ ED ρ ). When D is calcuwater (DW) and dose-to-medium σ ED andρ (D ee M M ρ ee σσED ρρ and ED ρ ρ lated, σσED and are based on the of e ED σ ED and ρ material properties e e ee voxel, i. When D is calculated, σσED ρ ρ ED output grid and and are σσED and W ED and and ρρ based on water. Since Equation 15 is calculated as an internal post processing operation after the energy dependent electron fluence is solved, both DM and DW can be theoretically obtained from a single transport calculation.

Spatial transport cutoff Acuros XB employs a spatial cutoff for photon energies below 1 keV and electron energies below 500 keV. When a particle passes below the cutoff energy, any subsequent interactions are assumed to happen locally in that voxel. Additional errors may also be present from the internally set convergence tolerances in Acuros XB. These tolerances control how tightly the inner iterations in Acuros XB are converged in energy group. These errors will generally be on the order of 0.1% of the local dose in any voxel. Page | 25 Page Page || 25 25

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Intended Use Summary The Eclipse™ treatment planning system (Eclipse TPS) is used to plan radiotherapy treatments for patients with malignant or benign diseases. Eclipse TPS is used to plan external beam irradiation with photon, electron and proton beams, as well as for brachytherapy treatments. In addition, the Eclipse Proton Eye algorithm is specifically indicated for planning proton treatment of neoplasms of the eye. Eclipse should only be used by qualified medical professionals.

Important Safety Information Radiation treatments may cause side effects, which, in some cases, may be serious. Severity can vary depending on the part of the body being treated. Side effects are related to the type of treatments delivered and should be discussed between the clinician and the patient.

Medical Advice Disclaimer Varian as a medical device manufacturer cannot and does not recommend specific treatment approaches. Individual treatment results may vary. © 2010, 2015 Varian Medical Systems, Inc. All rights reserved. Varian, Varian Medical Systems, Acuros, and RapidArc are registered trademarks, and Eclipse is a trademark of Varian Medical Systems, Inc. RAD 10156A

04/2015