## Highlights

- •Some biological model is needed to establish radiation therapy with helium-ions.
- •A biological model used for carbon-ion radiotherapy was investigated.
- •A ridge filter was designed and fabricated using the model to form a biologically uniform spread-out Bragg peak.
- •Biological experiments were performed in the spread-out Bragg peak.
- •It was found that the model could be applicable by some modification.

## Abstract

### Purpose

### Methods

### Results

### Conclusion

## Keywords

## 1. Introduction

Press Release: Heidelberg Ion Beam Therapy Center treats first patient using helium ion therapy with RayStation. https://www.raysearchlabs.com/media/press-releases/2021/heidelberg-ion-beam-therapy-center-treats-first-patient-using-helium-ion-therapy-with-raystation/, accessed February 15, 2023.

## 2. Materials and methods

### 2.1 MKM

where ${\alpha}_{0}$ and $\beta $ are parameters in the linear–quadratic (LQ) model in the limit LET = 0 and $D$ is the absorbed dose. The saturation-corrected dose-mean specific energy in a single event, ${z}_{1D}^{\ast}$, is given as [

where ${f}_{1}\left(z\right)$ is the probability density of $z$ deposited by a single energy-deposition event of a domain (the subnuclear volume defined in the MKM), and ${z}_{\mathit{sat}}$ is the saturation-corrected specific energy, expressed as

${z}_{0}$ is the saturation coefficient, expressed as

where ${R}_{n}$ and ${r}_{d}$ are the radii of the cell nucleus and the domain, respectively.

### 2.2 Formation of an SOBP

*k*

^{th}Bragg curve and ${D}_{k}(x)$ is the depth–dose curve of the

*k*

^{th}pristine Bragg curve for which a beam passes through a certain thickness of the range modulator, ${t}_{k}$. ${D}_{k}(x)$ is written as

where $I(x,c\bullet {t}_{k})$ is the planar integrated depth–dose curve in water (IDD) of a helium-ion beam and $c$ is the factor that converts the physical thickness to water-equivalent thickness, which is obtained from the ratio of the material stopping power to that of water. The IDDs for various thicknesses of the range modulator were obtained from $I(x,0)$, which was calculated using a Monte Carlo simulation toolkit, PTSim [

where ${L}_{\mathit{wx}}$ and ${L}_{\mathit{wy}}$ are the distances from the wobbling

*x*-magnet and

*y*-magnet to the isocenter, respectively. ${L}_{s}$ is the distance from the surface of the water phantom used for the dose profile measurements to the isocenter. This correction was necessary when the water phantom was placed at a fixed position during measurement. The last term of Eq. (5) represents the reduction of helium ions owing to inelastic nuclear interactions. The range modulator used in this study is made of aluminum, and $\mu $ is the attenuation coefficient of aluminum, which is calculated as 3.93 × 10

^{-3}mm using the inelastic cross sections of helium ions to aluminum. The cross-section of 652 mb at an energy of 210 MeV/u was obtained from Tripathi99 parameterization [

Total nuclear reaction cross-section database 2021. https://www.gsi.de/work/forschung/biophysik/fragmentation Accessed December 11, 2021.

where ${d}_{i}$ is the dose of particle $i$ at a depth of $x$. The dose-mean specific energy of the SOBP at depth $x$, ${\mathrm{Z}}_{\mathit{SOBP}}(x)$, is calculated as follows:

*k*

^{th}dose-mean specific energy, $Z(x,{c\bullet t}_{k})$, was obtained by inferring from $Z(x,0)$ (see the Appendix). The survival rate of the SOBP was calculated using Eq. (1), where $\alpha (x)$ and $\beta \left(x\right)$ at depth $x$ are given by:

The RBE is then calculated by ${D}_{\gamma}/{D}_{\mathit{He}}$, where ${D}_{\gamma}$ is the dose of a reference radiation, which is usually a photon, at a cell survival rate of $S$. The RBE-weighted dose at a depth of $x$ is then obtained by

Symbol | Quantity | Value |
---|---|---|

${\alpha}_{0}$[Gy^{−1}] | Linear parameter of the LQ model in the MKM | 0.172 |

$\beta $[Gy^{−2}] | Quadratic parameter of the LQ model in the MKM | 0.0615 |

${r}_{d}$[$\mathrm{\mu}$m] | Domain radius | 0.32 |

${R}_{n}$[$\mathrm{\mu}$m] | Nucleus radius | 3.9 |

### 2.3 Experimental apparatus and procedure

National Institute of Biomedical Innovation, Health and Nutrition 2021. https://www.nibiohn.go.jp/en/ Accessed November 30, 2021.

_{2}in the air). A 25 cm

^{2}plastic flask (Nalge Nunc International, Rochester, NY) on which cells in logarithmic growth were seeded was placed at the isocenter to be irradiated by the SOBP beam. A polyethylene (PE) block with a cross-sectional area of 200 × 200 mm

^{2}was placed in front of the flask to degrade the SOBP beam energy. Ten flasks were irradiated with the SOBP beam by changing the water-equivalent thickness of the block to 77, 117, 157, 177, 197, 217, 237, 257, 277, and 317 mm. The physical thickness of the PE block was calculated using the stopping-power table of ICRU49 [

^{3}, but it can vary from 0.94 to 0.97 g/cm

^{3}. The uncertainty of the stopping power can be up to 3%. The cells in each flask were trypsinized immediately after irradiation and plated on five plastic dishes (diameter: 60 mm) at the appropriate densities for clonogenic assays. Fixation with 10% formalin solution and staining with 1% methylene blue solution of colonies was performed after 14 days of incubation. Colonies consisting of more than 50 cells were considered viable. This experiment was independently performed three times on three different days.

## 3. Results

## 4. Discussion

^{−2}was consistent with the experimental results, as shown by the dashed line in Fig. 5. The adjustment was made by minimizing the variance between the measurements and the survival curve by fixing ${R}_{n}$ and ${R}_{d}$. The depth-cell survival curve of the SOBP was then recalculated using $\beta $ = 0.0457 Gy

^{−2}. The results are shown as dashed lines in Fig. 4. The cell survival rates over the SOBP region, recalculated using $\beta $ = 0.0457 Gy

^{−2}, were consistent with the measured values. The root mean square of the deviations between the measurement and expectation, with $\beta $ = 0.0457 Gy

^{−2}, was within 6.5%.

Symbol | Lee et al | Kase et al |
---|---|---|

${\alpha}_{0}$[Gy^{−1}] | 0.07 | 0.13 |

$\beta $[Gy^{−2}] | 0.0475 | 0.05 |

${r}_{d}$[$\mathrm{\mu}$m] | 0.43 | 0.42 |

${R}_{n}$[$\mathrm{\mu}$m] | 6.6 | 4.1 |

## 5. Conclusion

## Declaration of Competing Interest

## Acknowledgements

## Appendix A. Modeling of Bragg curves and dose-mean ${\mathit{z}}_{1\mathit{D}}^{\mathbf{\ast}}$ curves

^{3}was constructed by stacking 500 tallies with a thickness of 1 mm. Monoenergetic helium ions were impinged at the center of the phantom. The Bragg curve of the 210 MeV/u beam is shown in Fig. A1 with the measurements.

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