^{1,a)}, Joseph Perl

^{2}, Bruce Faddegon

^{3}, Jan Schümann

^{4}and Harald Paganetti

^{4}

### Abstract

**Purpose:**

To present the implementation and validation of a geometrical based variance reduction technique for the calculation of phase space data for proton therapy dose calculation.

**Methods:**

The treatment heads at the Francis H Burr Proton Therapy Center were modeled with a new Monte Carlo tool (TOPAS based on Geant4). For variance reduction purposes, two particle-splitting planes were implemented. First, the particles were split upstream of the second scatterer or at the second ionization chamber. Then, particles reaching another plane immediately upstream of the field specific aperture were split again. In each case, particles were split by a factor of 8. At the second ionization chamber and at the latter plane, the cylindrical symmetry of the proton beam was exploited to position the split particles at randomly spaced locations rotated around the beam axis. Phase space data in IAEA format were recorded at the treatment head exit and the computational efficiency was calculated. Depth–dose curves and beam profiles were analyzed. Dose distributions were compared for a voxelized water phantom for different treatment fields for both the reference and optimized simulations. In addition, dose in two patients was simulated with and without particle splitting to compare the efficiency and accuracy of the technique.

**Results:**

A normalized computational efficiency gain of a factor of 10–20.3 was reached for phase space calculations for the different treatment head options simulated. Depth–dose curves and beam profiles were in reasonable agreement with the simulation done without splitting: within 1% for depth–dose with an average difference of (0.2 ± 0.4)%, 1 standard deviation, and a 0.3% statistical uncertainty of the simulations in the high dose region; 1.6% for planar fluence with an average difference of (0.4 ± 0.5)% and a statistical uncertainty of 0.3% in the high fluence region. The percentage differences between dose distributions in water for simulations done with and without particle splitting were within the accepted clinical tolerance of 2%, with a 0.4% statistical uncertainty. For the two patient geometries considered, head and prostate, the efficiency gain was 20.9 and 14.7, respectively, with the percentages of voxels with gamma indices lower than unity 98.9% and 99.7%, respectively, using 2% and 2 mm criteria.

**Conclusions:**

The authors have implemented an efficient variance reduction technique with significant speed improvements for proton Monte Carlo simulations. The method can be transferred to other codes and other treatment heads.

J.R.M. acknowledges the Consejo Nacional de Ciencia y Tecnología (México) and H. Salazar-Ibargüen (Benemérita Universidad Autónoma de Puebla) for financial supporting for his visit to the Massachusetts General Hospital. J.R.M. thanks J. Shin (University of California San Francisco) for discussions on Geant4 topics. This work was supported by National Institutes of Health/National Cancer Institute (NIH/NCI) under R01 CA 140735-01.

I. INTRODUCTION

II. MATERIALS AND METHODS

II.A. The TOPAS code

II.B. The treatment head

II.C. Patient geometries

II.D. The geometrical particle splitting technique

II.E. Computational efficiency

III. RESULTS

III.A. Computational efficiency

III.B. Phase space analysis

III.C. Comparison of dose profiles

III.D. Patient calculations

III.D.1. Head treatment field

III.D.2. Prostate treatment field

IV. DISCUSSION

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Protons
- 38.0
- Monte Carlo methods
- 21.0
- Proton therapy
- 21.0
- Dosimetry
- 20.0
- Radiation therapy
- 12.0

## Figures

Treatment head at one gantry at the FHBPTC. The beam enters from the right. Dotted lines show the position of the split planes: upstream of the second scatterer (Sc2), at the second ionization chamber (IC2), and immediately upstream of the field specific aperture.

Treatment head at one gantry at the FHBPTC. The beam enters from the right. Dotted lines show the position of the split planes: upstream of the second scatterer (Sc2), at the second ionization chamber (IC2), and immediately upstream of the field specific aperture.

Normalized efficiency values for a single split plane at the second scatterer (Sc2), the second ionization chamber (IC2), and the aperture (AP) for options A1 and A8_1 (see Table I ). The contribution of secondary particles other than protons was discarded for the reference calculation of efficiency. Solid lines represent the fits using Eq. (4) .

Normalized efficiency values for a single split plane at the second scatterer (Sc2), the second ionization chamber (IC2), and the aperture (AP) for options A1 and A8_1 (see Table I ). The contribution of secondary particles other than protons was discarded for the reference calculation of efficiency. Solid lines represent the fits using Eq. (4) .

Normalized efficiency versus the number of splits for the two proposed settings. (Left) Setting 2: Upstream of the second scatterer (Sc2) and upstream of the aperture (Ap). (Right) Setting 1: Downstream of the second ionization chamber (IC2) and upstream of the aperture. Normalization was done with respect to the simulation without variance reduction. Five different geometrical setups (options; labeled as A1 to A8_2) were considered as described in Table I .

Normalized efficiency versus the number of splits for the two proposed settings. (Left) Setting 2: Upstream of the second scatterer (Sc2) and upstream of the aperture (Ap). (Right) Setting 1: Downstream of the second ionization chamber (IC2) and upstream of the aperture. Normalization was done with respect to the simulation without variance reduction. Five different geometrical setups (options; labeled as A1 to A8_2) were considered as described in Table I .

(Left) Effect on dose profiles by varying the split number for option A1. The difference from the reference curve (no splitting) is also shown, with the difference scale on the right side of the plot. (Right) Decrease in the efficiency due to the first split plane located upstream of the second scatterer (Sc2) rather than downstream of the second ionization chamber (IC2), with the second split plane located upstream of the aperture (Ap) (see Fig. 3 ).

(Left) Effect on dose profiles by varying the split number for option A1. The difference from the reference curve (no splitting) is also shown, with the difference scale on the right side of the plot. (Right) Decrease in the efficiency due to the first split plane located upstream of the second scatterer (Sc2) rather than downstream of the second ionization chamber (IC2), with the second split plane located upstream of the aperture (Ap) (see Fig. 3 ).

Planar fluence (left and top) and mean energy (right and top) per radial position for option A3 for the reference simulation (solid) and with variance reduction (dotted). The PHSP was divided into rings of equal area with a maximum radius of 5 cm to consider the penumbra of the beam. The dip at 4 cm in the mean energy is caused by the squared aperture (8 cm side). Relative differences in percent are shown at the bottom for both figures.

Planar fluence (left and top) and mean energy (right and top) per radial position for option A3 for the reference simulation (solid) and with variance reduction (dotted). The PHSP was divided into rings of equal area with a maximum radius of 5 cm to consider the penumbra of the beam. The dip at 4 cm in the mean energy is caused by the squared aperture (8 cm side). Relative differences in percent are shown at the bottom for both figures.

Energy spectrum (left and top) and angular distribution (right and top) for option A3. Reference simulation (solid) and simulation with variance reduction (dotted) are shown. Their relative differences in percent are shown at the bottom of the plots.

Energy spectrum (left and top) and angular distribution (right and top) for option A3. Reference simulation (solid) and simulation with variance reduction (dotted) are shown. Their relative differences in percent are shown at the bottom of the plots.

Depth–dose profile (left) and lateral dose profile at 5.5 cm and at 1 cm from the entrance of the water phantom for option A3. Percentage differences also are shown on the right axis.

Depth–dose profile (left) and lateral dose profile at 5.5 cm and at 1 cm from the entrance of the water phantom for option A3. Percentage differences also are shown on the right axis.

Depth–dose profile at depth (left) and lateral dose profile (right) at 23 cm and at 10 cm from the entrance of the water phantom for option A8_1. Percentage differences below of 2% also are shown on the right axis.

Depth–dose profile at depth (left) and lateral dose profile (right) at 23 cm and at 10 cm from the entrance of the water phantom for option A8_1. Percentage differences below of 2% also are shown on the right axis.

Transverse view for a head treatment. Reference simulation (solid) and with variance reduction (dotted) are shown in the same image. The right side shows the gamma test values. The percentage of total voxels with a gamma value lower than unity is 98.9% by using a 2 mm and 2% criteria.

Transverse view for a head treatment. Reference simulation (solid) and with variance reduction (dotted) are shown in the same image. The right side shows the gamma test values. The percentage of total voxels with a gamma value lower than unity is 98.9% by using a 2 mm and 2% criteria.

Coronal view for a prostate case. (Left) Reference simulation (solid) and with variance reduction (dotted) are shown. The right side shows the gamma test values. The percentage of total voxels with a gamma value lower than unity is 99.7% by using a 2 mm and 2% criteria.

Coronal view for a prostate case. (Left) Reference simulation (solid) and with variance reduction (dotted) are shown. The right side shows the gamma test values. The percentage of total voxels with a gamma value lower than unity is 99.7% by using a 2 mm and 2% criteria.

## Tables

Proton beam configuration options used in the study for calculating dose distributions in water. These options cover the minimum and maximum proton ranges deliverable at the MGH gantry treatment heads.

Proton beam configuration options used in the study for calculating dose distributions in water. These options cover the minimum and maximum proton ranges deliverable at the MGH gantry treatment heads.

Average simulation times per CPU, efficiency and normalized efficiency per CPU for the reference simulations and the simulations using variance reduction for the production of PHSP. The normalization was made with respect to the reference simulations. The planar energy fluence from a bin of 1 cm radius was considered to calculate the variance. The statistical uncertainty of the full PHSPs is on average lower than 0.2% for all options.

Average simulation times per CPU, efficiency and normalized efficiency per CPU for the reference simulations and the simulations using variance reduction for the production of PHSP. The normalization was made with respect to the reference simulations. The planar energy fluence from a bin of 1 cm radius was considered to calculate the variance. The statistical uncertainty of the full PHSPs is on average lower than 0.2% for all options.

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