Generalized eMC implementation for Monte Carlo dose calculation of electron beams from different machine types

The new eMC beam model for electron beams using applicators up to 25 × 25 cm² consists of 4 sub-sources illustrated in figure 1: a 'main electron source' and a 'main photon source' representing electrons and photons coming from the scattering foil (sub-source 1 in figure 1), an 'edge source of electrons' which accounts for electrons produced at the edges of the scrapers of the applicator or insert (sub-source 2 in figure 1), a 'source of transmitted photons' through the last applicator scraper or insert (sub-source 3 in figure 1) and a 'line source' to model the head scatter radiation of electrons and photons (sub-source 4 in figure 1).

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For the commissioning of the electron beam model the following set of measurements are needed with gantry angle and collimator angle set to zero. SSD and source to detector distance (SDD) are measured from a nominal source position 100 cm above the isocenter of the accelerator:

• In-air cross-line and in-line dose profiles p_x(x) and p_y(y), respectively, with secondary collimator jaws at 40 × 40 cm² at an SDD of 95 cm (one each per energy without applicator).
• Depth dose curves in units of cGy/MU in water at a SSD of 100 cm, with secondary collimator jaws set to 40 × 40 cm² (one per energy without applicator).
• In-air cross-line and in-line dose profiles a_x(x) and a_y(y), respectively, with secondary collimator jaws at field sizes used for the different applicators at an SDD of 95 cm (one each per energy and jaw setting, but measured without applicator).
• Depth dose curves in units of cGy/MU in water at an SSD of 100 cm (one per energy and applicator).

The 'main electron source' and the 'main photon source' are point sources located at 90 cm above the isocenter, approximately at the level of the primary scattering foil. The starting point of electron histories for the dose calculation is sampled from a two-dimensional fluence function at an SSD of 95 cm. This fluence function is determined by the two sets of measured in-air profiles p_x(x), p_y(y), a_x(x) and a_y(y). The measured profiles p_x(x), p_y(y) with secondary collimator jaws set to 40 × 40 cm² are used to determine the radial function f_r(r) of the fluence:

$$\begin{equation} f_r ( r ) = \frac{{p_x (r) + p_x ( - r) + p_y (r) + p_y ( - r)}}{4}.\end{equation} \tag{ 1 }$$

Since the width of these profiles is larger than the length of the diagonals for the largest applicator used (25 × 25 cm²), this radial function covers the fluence in the field corners. The applicator-specific in-air profiles a_x(x) and a_y(y) are used to take into account the specific characteristics of the jaw settings on the fluence:

$$\begin{equation} g_x ( x ) = \frac{{a_x (x)}}{{f_r (| x |)}} \quad{\rm and}\quad g_y ( y ) = \frac{{a_y (y)}}{{f_r (| y |)}}. \end{equation} \tag{ 2 }$$

Finally the two-dimensional fluence function f(x,y) is then

$$\begin{equation} f( {x,y} ) = f_r \left( {\sqrt {x^2 + y^2 } } \right) g_x ( x )g_y ( y ). \end{equation} \tag{ 3 }$$

The range of the variables x and y are given by the measurements. The direction of the starting particle is defined by the line from the point source to the starting point for photons and electrons. For the electrons this direction has to be corrected for the in-air scatter between the source and the starting point at an SSD of 95 cm. The correction is carried out by sampling from a Gaussian distribution for the direction cosines u and v with a mean equal to zero and a sigma which is determined analytically using a small angle approximation for multiple scattering in air. The sigma is a function of the energy of the electron σ_θ(E) (ICRU 1984).

BEAMnrc simulations (Rogers et al 1995) are performed for each electron beam energy to determine the two-dimensional fluence at an SSD of 95 cm and the energy spectrum of the main photon source. For this purpose the main accelerator head components, i.e. primary collimator, vacuum window, scattering foils, monitor chamber, mirror, secondary collimator jaws and reticle, are implemented into the BEAMnrc package and phase space files at an SSD of 95 cm are generated. The energy spectrum of the main electron source is determined by de-convolving the measured depth dose curve in water at an SSD of 100 cm, with secondary collimator jaws set to 40 × 40 cm². For this purpose pre-calculated mono-energetic macro MC depth dose curves for the main electron source without applicator and a 40 × 40 cm² field size are used together with the pre-calculated photon depth dose curve of the main photon source. Besides the energy spectrum of the electrons from the main electron source, the source weights of the main electron and main photon source are the result of this de-convolution.

The 'edge source of electrons' accounts for electrons produced at the edges of the upper and middle scraper and lowest applicator scraper or insert. This source consists of two parts. The first part is a line source along the edge of the scrapers or inserts top face and accounts for electrons coming from the inner side of the scrapers or insert, which are produced by electrons of the main electron source entering the material on the top face. The second part covers electrons coming from the inner side of the scrapers or insert which are produced by electrons of the main electron source impinging the material on the side. Since the fluence contribution depends on the electron fluence of the main electron source, the source weight of this sub-source is proportional to the main electron source weight. The electrons from this edge source are sampled by making use of EGSnrc pre-calculated scatter kernels. These kernels are determined by simulating the radiation transport of electrons impinging on the edge of the scraper for a variety of energies and directions of the incoming electrons. The direction and the energy of the electrons leaving the inner side of the scraper are scored. That is, the parameters of the kernel are the incoming electron energy and direction and the results of the sampling are the outgoing energy and direction of the electron. The orientation of the kernel is adjusted according to the incoming position and direction of the electron and the orientation of the scrapers or inserts (divergent or non-divergent).

The 'transmitted photon source' is located on the lower plane of the last applicator scraper or insert and consists of three parts. The first part includes scattered photons produced by electrons of the main electron source in the scraper or insert. These photons are sampled by making use of pre-calculated scatter kernels using EGSnrc. Analogous to the above described kernel for the edge source, the parameters of this kernel are the incoming electron energy and direction and the results of the sampling are the outgoing energy and direction of the photon. The second part of the transmitted photon source covers the photons which pass through the material without any interaction. Thus, they have the same direction as the photons form the main photon source but the energy is sampled from pre-calculated transmission spectra using EGSnrc. The third part takes into account the scattered photons produced by the photons form the main photon source in the material. The scattered photons are sampled by use of pre-calculated scatter kernels using EGSnrc with the incoming photon energy as input and the photon energy and direction of the scattered photon as output. The orientation of the kernels used in this sub-source is adjusted according to the incoming direction of the particle. Since the fluence contribution depends on the fluence of the main electron and photon source, the source weights of this sub-source are proportional to the weight of the main electron and photon source.

The 'line source' represents the head scatter. For each secondary collimator jaw a line source for electrons and photons is defined. The origin of the line source is along the horizontal line segment located at the center of the inner side of the jaws. The width of this segment corresponds to the jaw setting applied for the applicator considered. The location of the origin point is sampled uniformly. The starting point of the particle is sampled in a plane at 95 cm (i.e. 5 cm above the iso-center plane) inside the field defined by the applicator or the insert. The mean direction of the electrons and the direction of the photons are given by the sampled starting position and the sampled origin position. The direction of the electrons is then varied according to a Gaussian distribution with an energy dependent sigma σ_θ(E) analog to the main electron source. The relative weight of the particles from the individual collimator jaws are determined by analyzing phase space files of BEAMnrc simulations of the accelerator head. As MC dose calculations showed a dose contribution of about 10% from the secondary collimator jaws to the total dose, the weight of the line sources is chosen to also achieve a 10% dose contribution from the line source to the total dose. This approach is applied to all linear accelerators, beam energies and applicators. The energy spectrum for the line photon source is taken from the corresponding BEAMnrc simulation for the electron beam energy considered. The energy spectrum of the electron line source is determined using the measured depth dose curve in water at an SSD of 100 cm for the applicator considered. For this purpose pre-calculated mono-energetic macro MC depth dose curves for the line electron source with applicator and the pre-calculated photon depth dose curve of the line photon source are used. Since the measured depth dose includes the contribution of all sources, first the dose contribution assigned to the line source has to be determined. This is achieved by calculating the depth dose curve using all sub-sources except the line source for each energy and applicator combination. The resulting calculated depth dose is subtracted from the measured depth dose curve for the energy and applicator considered. The remaining part of the measured depth dose is used to determine the energy spectra of the electron line source by de-convolving as for the main electron source.

Since the two-dimensional fluence function is based on in-air measured profiles without applicator in place, the sampling of the starting point for the main electron source and the line sub-source for a specific energy-applicator combination does not take into account the impact of the upper and middle scrapers of the applicator. This leads to significant dosimetric errors in the dose profiles. The idea is to correct for this by back projecting the electrons to the plane of the scrapers and determine if the electrons pass the opening of the scrapers or not (first for the middle scraper and second for the upper scraper). If they pass, the transport will be continued; otherwise the particle will be rejected. This procedure is described as follows: Two independent 4D probability distributions are determined using the small angle approximation for the in-air multiple scattering and a uniform fluence distribution in the plane at an SSD of 95 cm. The in-air multiple scatter in the x-direction is described by the distribution of the four variables x_in, u_in, x_out and u_out, where x_in and u_in are the x-position and x-direction cosine in a given plane (input plane), e.g. at an SSD of 95 cm, and x_out and u_out are the estimated x-position and x-direction cosine in another plane (output plane), e.g. the plane where the particle passes the middle scraper. The second probability distribution is considering the y-position and the y-direction and is a function of y_in, v_in, y_out and v_out. These distributions are now applied: Given a starting point of the electron (x₉₅, y₉₅) and the corresponding starting direction cosines (u₉₅, v₉₅) at an SSD of 95 cm, the point (x_m, y_m) and direction (u_m, v_m) in the plane of the middle scraper can be determined by sampling from the 2D Gaussian distributions for (x_m, u_m) and (y_m, v_m), respectively. The electron is rejected if it does not pass through the opening of the middle scraper. Otherwise the point (x_u, y_u) and direction (u_u, v_u) in the plane of the upper scraper is determined analogously based on (x_m, y_m) and (u_m, v_m) as input. The electron is rejected if it does not pass through the upper scraper's opening. The method of accounting for the in-air multiple scatter is a more general implementation than the method described in a previous publication (Fix et al 2010).

This new beam model differs considerably from the previously one. It includes edge sources of electrons for all scrapers of the applicator. The second point source of the previous beam model is replaced by the line sources to represent the head scatter more accurately. The new beam model takes the air scatter of electrons in the applicator into account. The fluence distribution of the main electron source in the new beam model is based on more in-air profiles which allows for more accurate modeling of the fluence compared to the previous model where just one in-air profile is used.

The modifications developed to improve the macro MC method for low electron beam energies as described by Fix et al (2010) were included in the newly developed transport code which is now implemented in C++. The particle transport of the primary electrons is not changed, except that a higher resolution for the PDFs generated in the local MC simulations is used. The electron with the largest energy leaving the sphere is defined as the exiting primary electron. The sampled energy loss is deposited along a straight line between the entrance and the sampled exit position of the primary electron for the current step. The energy deposition of the primary electron $E_{{\rm dep}_i }^p$ in a voxel i along this path is determined according to

$$\begin{equation} E_{{\rm dep}_i }^p = \frac{{E_{{\rm dep}}^p }}{l} \cdot \Delta l_i \cdot {\rm SPR}_k^i \end{equation} \tag{ 4 }$$

where $E_{{\rm dep}}^p$ is the primary electron energy to be deposited in the current step, l is the total length of the current transport step, Δl_i is the length of the ray segment inside the voxel i and ${\rm SPR}_k^i$ is the stopping power ratio of the voxel material to the sphere material. More information about the primary electron transport is available in the Eclipse Algorithms Reference Guide (Varian, Varian Medical Systems 2008).

Compared to the original eMC implementation, the transport of secondary particles is modified considerably. The local MC simulations for spheres with diameters of 1, 2, 3, 4 and 6 mm are performed for different materials. In these local simulations, the number of secondary particles together with their direction and energy are scored as functions of the initial primary electron energy, the sphere size and the material. In most cases no secondary particles leave the spheres. To increase the efficiency of the macro MC, one secondary electron and one secondary photon are generated in each macro step. The energies of the secondary particles are sampled from the energy distributions of the secondary electrons and photons of the corresponding local simulations. To achieve energy conservation, on average, a particle weight is introduced. This weight for the secondary particle w(E) is determined by means of the probability that a secondary particle is created during the local simulation:

$$\begin{equation} w( E ) = \frac{{N_s (E)}}{{N_p (E)}} \end{equation} \tag{ 5 }$$

where N_s(E) is the number of secondary particles occurring in the local simulation of N_p(E) incident electrons of energy E. The direction of each secondary particle is sampled from the angular distribution as a function of the energy of this particle. The starting position of the secondary particle is located within the sphere and is sampled uniformly along the direction of the incident primary electron. The energy deposition of the secondary electron $E_{{\rm dep}}^s$ is scored along its direction and is determined for a voxel i according to

$$\begin{equation} E_{{\rm dep}_i }^s = w(E) \cdot 2MeV/{\rm cm} \cdot \Delta l_i \cdot {\rm SPR}_i \end{equation} \tag{ 6 }$$

where SPR_i is the stopping power ratio of the voxel's material to water.

For the secondary photon the distance to the next interaction is determined taking the local mass attenuation of the material into account and applying ray tracing from the sampled starting position along the sampled direction. At the interaction position the energy which is transferred from the photon to the electron is sampled from a PDF depending on the photon energy using data from XCOM (Berger et al 2010). The weight of the electron is set to the weight of the photon. Whereas the photon is assumed to leave the patient geometry, the energy of the electron is deposited along the photon's direction according to equation (6).

The macro MC is an adaptive step size algorithm, i.e. if the primary electron position is close to a heterogeneity, smaller sphere sizes are used for the steps to assign the correct material for the radiation transport. In homogeneous regions large sphere sizes are used. In previously implemented improvements, which are also included in this new version, the maximal sphere size used is reduced depending on the initial electron energy. This increases the accuracy for low electron beam energies. In this work the adaptive step size algorithm is modified such that large changes in the used sphere sizes are prevented, e.g. instead of switching from the largest to the smallest sphere size an intermediate sphere size is used.

In the first step, the improved macro MC implementation used as a dose calculation algorithm is tested. For this purpose, MC calculated dose distributions in homogeneous and inhomogeneous phantoms using either the macro MC transport or EGSnrc are compared. The EGSnrc calculated dose distributions are considered as benchmark. The homogeneous phantoms with a dimension of 40 × 40 × 20 cm³ consisted of the following materials: water, PMMA, lung (0.3 g cm⁻³) and bone (1.84 g cm⁻³). In the first inhomogeneous phantom with the same overall dimensions, a 1 cm thick water slab is followed by a 2 cm thick lung slab, a 1 cm thick bone slab and 16 cm of water. In the second inhomogeneous phantom a 2 cm thick air slab is used instead of the lung slab. The voxel size for the dose calculations is 2 × 2 × 1 mm² (1 mm in the incident beam direction), and the statistical uncertainty of the MC calculated dose distributions is less than 0.5% (1 std. dev.). The dose calculations are performed for different mono-energetic divergent electron beams (4, 6, 9, 12, 16, 22 and 25 MeV) with an SSD of 100 cm and a field size of 10 × 10 cm². The dose distributions are compared in dose per unit fluence, i.e. in units of Gy • cm². For the analysis of this comparison a 3-dimensional gamma evaluation is performed using EGSnrc as reference and the macro MC as evaluation distribution with 2%/2 mm and 1%/1 mm criteria (Low et al 1998). The dose difference criteria are relative to the maximum dose in the reference and a threshold equal to the dose difference criteria is used which discards voxels of dose values below this threshold for the gamma evaluation.

In the second step, the generalized eMC is validated, i.e. by considering commissioned beam models together with the macro MC dose calculation algorithm. During the commissioning the beam model parameters are configured leading to the machine-specific beam model which is then used for all dose calculations. Calculated and measured dose distributions are compared for different linear accelerators: Varian Clinac 2300C/D, Siemens Primus and Elekta Precise. For the Varian linac beam energies of 4, 6, 9, 12, 16 and 20 MeV with applicators 6 × 6, 10 × 10, 15 × 15, 20 × 20 and 25 × 25 cm², for the Siemens linac beam energies of 6, 9, 13, 17 and 21 MeV with applicators 10 × 10, 15 × 15, 20 × 20 and 25 × 25 cm² and for the Elekta linac beam energies of 6, 9, 12 and 15 MeV with applicators 6 × 6, 10 × 10, 14 × 14 and 20 × 20 cm² are available and no inserts are investigated. The measurements are performed using Si electron diodes (Scanditronix Wellhöfer, Schwarzenbruck, Germany) and a water tank. Besides the measurements needed for the configuration of all the energy-applicator combinations, several validation measurements are performed in a water phantom. These measurements consist of depth dose curves and lateral dose profiles for all available applicators at several SSDs ranging from 100 to 120 cm depending on the machine type. The lateral dose profiles are measured at several depths in water. The uncertainty of the diode measurements is estimated to be 1% (Zhang et al 1999). The corresponding dose calculations are performed in a 40 × 40 × 20 cm³ water phantom with voxel sizes of 4 × 4 × 1 mm³ (1 mm in incident beam direction). The dose profiles at a certain depth of measurement, i.e. a specific z-position, are determined by linear interpolation of the nearest neighbor's dose profiles in the z-direction. No smoothing is applied to the results of the dose calculations. The statistical uncertainty of the MC calculated dose distributions is less than 1% (1 std. dev.).

In the first test the depth dose curves calculated with either EGSnrc or macro MC are compared for each combination of 4 materials (water, PMMA, lung, bone) and 7 energies (4, 6, 9, 12, 16, 22 and 25 MeV) for a field size of 10×10 cm². As examples, in figure 2, the results of the comparisons between the calculated depth dose curves are shown for different mono-energetic beam energies and in different homogeneous materials. Dose differences are stated in% relative to the maximum of the dose distribution calculated with EGSnrc for all comparisons between EGSnrc and eMC. The gamma evaluation shows that more than 99% and more than 95% of the analyzed voxels have a gamma <1 for the 2%/2 mm and the 1%/1 mm criteria, respectively. However, an exception is the dose comparison for 4 MeV in lung where dose differences up to 3% occurred in the depth dose curve and the number of voxels passing the gamma criteria of 2%/2 mm and 1%/1 mm is reduced to 98% and 93%, respectively.

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The second test considered two water phantoms with lung-bone and air-bone slabs as inhomogeneities and a field size of 10 × 10 cm². Dose distributions for the 7 mono-energetic electron beams are calculated using EGSnrc or the macro MC method for both phantoms. Figure 3 shows examples for the resulting depth dose curves. Macro MC dose values follow those calculated with EGSnrc at the material interfaces. The gamma evaluation for these inhomogeneous phantoms results in a percentage of voxels passing the gamma criteria of 2%/2 mm and 1%/1 mm to be 99% and 94%, respectively. The results in this section demonstrate the accuracy of the macro MC transport for dose calculations.

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In this section the calculated dose distributions using eMC with the electron beam model are compared with measurements in water for linear accelerators of different manufacturers. All dose comparisons are presented in units of cGy per MU. All dose differences shown are relative to the maximum dose of the measurement. In figure 4 the comparisons of measured and calculated depth dose curves including dose difference curves are illustrated for a Varian linear accelerator for two different energy-applicator combinations and two different SSDs. The agreement is within 2% or 1 mm for all the energy-applicator combinations of the Varian machine for SSDs between 100 and 120 cm. Comparisons of lateral dose profiles for this type of linear accelerator are illustrated in figure 5 for two settings of energy-applicator combination and SSDs. The depth of 3.2 cm and 2.2 cm corresponds to the depth of maximum dose and the depth of 6.8 cm and 3.7 cm to R₅₀, respectively. An agreement within 1% or 1 mm is found for the comparison of the dose profiles.

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Figures 6 and 7 provide examples of the comparison of calculated and measured dose distributions for Elekta machines. Generally, the agreement between calculated and measured depth dose curves is within 2% or 1 mm for all energy-applicator combinations considered. Figure 7 depicts the comparison of the available measured lateral dose profiles in water with the corresponding calculated dose values at depths of maximum dose and 10 cm in water. Overall, all the calculated and measured dose values agree within 2% or 2 mm. The agreement for Elekta machines is generally the best at SSD = 100 cm and decreases for extended SSDs.

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Finally, the dose comparisons are performed for a Siemens linear accelerator. Figures 8(a) and (b) show the comparisons of two energy-applicator combinations at SSDs of 100 cm and 115 cm, respectively. The agreement between the calculated and measured depth dose curve is within 2% or 1 mm for the beam-applicator combinations considered. In figure 8(b) there is a systematic difference of about 1% in the high dose region. The dose comparisons between calculated and measured dose profiles are shown in figure 9 for two energy-applicator combinations at an SSD of 100 cm and 115 cm at several depths in water. An overall agreement between calculated and measured dose curves for Siemens machines is found to be within 2% or 2 mm.

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