## Highlights

- •Pencil Beam Redefination Algorithm (PBRA) was implemented with three initialization methods and compared to MC.
- •Phase space distribution of the electrons below the applicator was generated using MC simulation.
- •The agreement between PBRA dose calculation and MC is promising for development of an intraoperative TPS for IOERT.

## Abstract

### Purpose

### Methods

^{2}pixels and energy bin width of 1 MeV.

### Results

### Conclusions

## Keywords

## Introduction

## Materials and methods

### Theory

*σθ*) was assumed independent of position based on the methodology proposed by Hogstrom et al. [

*σθ*formula depended on the depth, linear angular scattering power, and source–secondary collimator distance (SCD) [

### PBRA initial parameters

#### The input dataset

^{2}pixels. The energy was considered discrete with 1 MeV bins. The planar fluence was equal to the number of electrons in each pixel and energy bin. Direction of electrons should be calculated in both X_Z and Y_Z planes, separately. According to Shiu et al. [

*Z*axis (

*φ*angle in Fig. 1). Determination of this angle from the output data from the PTRAC card was done based on the following equations. The initial directions in the Y_Z plane are the transpose of the X_Z plane’s directions, according to symmetry.

where

*u*,

*v,*and

*w*are cosine of the crossing particle directions, given by the PTRAC card, with

*X*,

*Y,*and

*Z*axes, respectively.

#### Determination of direction parameters

*Y*axis were combined because of system’s symmetry, and the fits with less than 10 data points were ignored.

- 1)Parameters extracted from the PSD data using the PTRAC card (PTRAC).
- 2)Fitting formulas for the mean and SD angles presented in section 3.2 (Formula).
- 3)The method proposed by Hogstrom et al. [[13]] (Hogstrom).

*X*–

*Z*plane was calculated based on the beam divergence equal to arctan (

*x*/SSD), where SSD stands for the geometric source–surface distance in LIAC that is usually equal to 713 mm [

*Y*-

*Z*plane as arctan (

*y*/SSD). SD angles were calculated by equations presented by Hogstrom et al. [

^{3}dimension with a mesh dimension of 2 × 2 × 5 mm

^{3}was simulated in both MCNP and PBRA, and the recorded energy depositions were compared in terms of gamma index.

### Implementation of PBRA algorithm

#### Determination of correction factors

*C*(

*E*)) plays a crucial role in the accuracy of PBRA dose calculation. It accounts for the neglected physics of electron interactions (e.g., small-angle scattering approximation) [

*C*(

*E*) determination, i.e. fitting a high-order polynomial to the energy dependent correction factor values. The gold standard in this study was the dose distribution in the water phantom calculated by the validated MC. The electrons were transported while the applicator was placed perpendicular to the phantom surface and the PDD curve was calculated using the energy deposition tally. The absorbed doses were recorded in 2 × 2 × 5 mm

^{3}cells. The number of histories was enough to achieve less than 3 % uncertainty. The degree of polynomial function fitted to the correction factors versus energy was chosen in such a way to obtain the least possible dose difference between the PBRA and MC simulation.

#### Consideration of the backscatter of the protecting disk

Baghani HR. Possibility evaluation of intraoperative imaging, treatment planning and simultaneous dosimetry during breast cancer intraoperative radiotherapy and comparing the results of preoperative and postoperative treatment planning in this treatment technique. Iran: Shahid Beheshti University; 2014.

where ${\text{D}}_{\text{disk}}$ is the dose in the presence of the protecting disk and ${\text{D}}_{\text{water}}$ is the dose without the disk.

#### PBRA parallelization

^{2}field divided by 2 × 2 mm

^{2}pixels, the number of threads was 51

^{4}. A unique combination of 4 numbers from 0 to 50 should be extracted from each thread number to specify the X and Y coordinates of the current and previous depth pixels. In other words, the number should be converted from base 10 to base 51, and each digit represents the required number. Fig. 2 shows the number of thread which is responsible for calculating the deposition energy of the electron which is traveling from pixel (1,49) in the previous depth to pixel (50,0) in the current depth.

### Validation of PBRA

#### Validation of PBRA in a water phantom

^{3}cubic water phantom was used with the applicator perpendicular to the phantom surface. The energy deposition was recorded in voxels with the size of PBRA voxels (i.e., 2 × 2 × 5 mm

^{3}) with the largest dimension in the depth direction (

*Z*axis). Comparison between the PBRA results and MC simulation was performed in terms of gamma index with criteria of 3 % dose difference (DD) and 3 mm distance to agreement (DTA) [

#### Validation of PBRA in a mathematical phantom

Breast Size Chart. February 2022. https://www.averageheight.co/breast-cup-size-by-country.

Bra-size by country of origin. February 2022. https://www.worlddata.info/average-breastsize.php.

Baghani HR. Possibility evaluation of intraoperative imaging, treatment planning and simultaneous dosimetry during breast cancer intraoperative radiotherapy and comparing the results of preoperative and postoperative treatment planning in this treatment technique. Iran: Shahid Beheshti University; 2014.

^{3}was used to calculate the energy deposition. The same voxel size, energy and geometry were used in PBRA with formula-based initialization. To consider the back scatter of the protecting disk in the PBRA, the method described in section 2.3.2 was used. Comparison between the dose distributions of PBRA and MCNP was made in terms of gamma index with criteria of 3 % DD and 3 mm DTA.

## Results

### Determination of direction parameters

*R*

^{2}) of the fits for energies higher than 3 MeV was higher than 0.9 and, for all energies, it was higher than 0.7. The lower

*R*

^{2}value for lower energies was not crucial due to their negligible electron fluence.

### Determination of the mean and SD angle formulas

where

*a*and

*b*are the fitting parameters and

*×*is the X coordinate of the pixels relative to the central axis. The rightmost term in equation (2) shows the convergence of the curves. It was one of the fitting parameters, which resulted in a value close to

*E*

_{0}/2 (

*E*

_{0}is the nominal energy of the accelerator). By replacing the parameter with

*E*

_{0}/2, the 95 % confidence intervals (CI) did not change considerably and even improved in some cases. The parameter values as well as their 95 % confidence intervals (CI) for different

*E*

_{0}are presented in Table 1.

*R*

^{2}values for different nominal energies of LIAC.

Nominal Energy | Equation 2 | Equation 3 | ||||||
---|---|---|---|---|---|---|---|---|

a | 95 % CI | b | 95 % CI | ${R}^{2}$ | A | 95 % CI | ${R}^{2}$ | |

6 MeV | 0.7206 | 0.6591, 0.7822 | 0.0322 | 0.0278, 0.0368 | 0.9694 | 0.01679 | 0.0164, 0.0172 | 0.953 |

8 MeV | 0.7610 | 0.7092, 0.8128 | 0.0232 | 0.0210, 0.0254 | 0.9871 | 0.0145 | 0.0143, 0.0147 | 0.9801 |

10 MeV | 1.0490 | 0.9137, 1.1840 | 0.0100 | 0.0086, 0.0114 | 0.9956 | 0.0101 | 0.0100, 0.0101 | 0.9952 |

12 MeV | 1.2600 | 1.0810, 1.4380 | 0.0095 | 0.0081, 0.0110 | 0.9955 | 0.01154 | 0.0115, 0.0116 | 0.9952 |

*x*. This provides the advantage of more straightforward implementation and less fitting parameters.

*A*, as well as goodness of fit values are presented in Table 1.

where

*a*and

*b*are fitting parameters. The fitting parameters and their 95 % CI for different

*E*

_{0}values are given in Table 2.

*R*

^{2}values for different nominal energies of LIAC.

Nominal Energy | a | 95 % CI | b | 95 % CI | ${\mathit{R}}^{2}$ |
---|---|---|---|---|---|

6 MeV | 63.61 | 56.30, 70.92 | 0.3735 | 0.3354, 0.4116 | 0.9926 |

8 MeV | 61.79 | 56.41, 67.17 | 0.2865 | 0.2641, 0.3089 | 0.9909 |

10 MeV | 60.77 | 56.63, 64.92 | 0.2439 | 0.2275, 0.2603 | 0.9935 |

12 MeV | 62.30 | 57.05, 67.55 | 0.2404 | 0.2194, 0.2613 | 0.9878 |

### Comparison of different initial parameters

Nominal Energies | Formula | PTRAC | Hogstrom |
---|---|---|---|

6 MeV | 99.3 % | 99.7 % | 86.4 % |

8 MeV | 99.9 % | 98.5 % | 97.1 % |

10 MeV | 99.7 % | 98.2 % | 96.8 % |

12 MeV | 99.9 % | 99.9 % | 96.5 % |

### Validation of the PBRA implementation

#### Validation of the PBRA in a water phantom

^{3}. The obtained gamma values are reported in the whole electron beam FOV and up to the last nonzero depth dose in PBRA (Fig. 8). The voxels outside the FOV were not taken into account where MC error was not acceptable. The 3D gamma passing rates for 6, 8, 10, and 12 MeV were 99.3 %, 99.9 %, 99.7 %, and 99.9 %, respectively.

#### Validation of the PBRA in a mathematical phantom

## Discussion

### Mean and SD of deflection angles

### Different initialization methods

### PBRA validation

^{3}water phantom with a grid size of 2 × 2 × 5 mm

^{3}. The time consumed in the presence of inhomogeneities will not be changed considerably because neither the number of threads working simultaneously, nor the number of variables which should be calculated in a thread will be changed. Calculation time depends on the field size and the depth at which the electron fluence becomes zero (e.g. after the depth of shielding disk or the energy dependent electron range).

## Conclusion

## Declaration of Competing Interest

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