## Highlights

- •Uncertainties in proton therapy beam delivery.
- •Uncertainty Quantification methods in beam dynamics.
- •Analysis of errors in transport lines and their effects on the beam properties.
- •Optimized beam optics solutions using UQ methods.

## Abstract

## Keywords

## 1. Introduction

## 2. Uncertainty Quantification theory

*u*is described as a series expansion of basis vectors ${\mathrm{\Psi}}_{i}$ , function of input parameters $\mathit{\xi}$ with corresponding coefficients ${\alpha}_{i}$ as:

where $\mathit{\xi}=({x}_{1},{x}_{2},\dots ,{x}_{d})$ is the vector of input parameters ${\mathrm{x}}_{j}$ (with

*j*= 1, $\dots $ ,

*d*) that represent possible sources of uncertainty. Due to practical limitations, the PCE series in Eq. (1) is truncated up to a certain order

*p*.

### 2.1 Training points

*p*of the surrogate model to be built and the number

*d*of input parameters as:

### 2.2 PCE basis

*p*, the PCE surrogate model contains only a limited number of basis vectors represented by a sequence of orthogonal polynomials. According to the Askey scheme [

Distribution | Polynomial | Support |
---|---|---|

Gauss | Hermite | [−$\infty ,\infty $] |

Uniform | Legendre | [−1,1] |

Exponential | Laguerre | [0,$\infty $] |

Beta | Jacobi | [−1,1] |

*p*with

*d*input parameters, it is necessary to introduce the multi-indices $\mathit{M}$ = (${m}_{1},$…$,{m}_{d}$) of degree

where ${\psi}_{{m}_{k}}^{k}({x}_{k})$ is, in our case, the univariate Legendre polynomial of degree ${m}_{k}$ associated to the input ${x}_{k}$.

### 2.3 Coefficients

*p*of the PCE and on the number

*d*of input parameters as:

where the

*i*-th coefficient is obtained by the orthogonal projection of

*u*onto the

*i*-th basis polynomial. The integral at nominator in Eq. (6) is evaluated numerically by quadrature rules [

where ${\delta}_{\mathit{ij}}$ denotes the Kronecker delta.

### 2.4 Statistical information

where the t-Student parameter is adjusted with the number

*d*of input parameters [

### 2.5 Sensitivity analysis

where ${\mathbb{I}}_{i}^{s}$ is the set of multi-indices ${m}_{k}$ that include

*i*-th order only. In principle a large value of the index ${S}_{i}$ means that the associated parameter ${x}_{i}$ is important in the variation of a certain QoI.

### 2.6 Setup of UQ analysis

Adelmann A, et al. The OPAL (Object Oriented Parallel Accelerator Library) Framework. Technical Report PSI-PR-08-02, Paul Scherrer Institut, 2008-2019.https://gitlab.psi.ch/OPAL/src/wikis/home.

*p*of the surrogate model. Depending on the number of input parameters, a low-order surrogate model (e.g.

*p*= 2) has the advantage to require only a few training points to be developed. However, in case of complex large-scale models, this may return in an inaccurate surrogate model and wrong response of the QoI variation. An example of the inaccuracy of low-order surrogate model is shown in Fig. 2 in comparison with the expected results of the high-fidelity model evaluated at the training points.

*p*= 4 is needed as displayed in Fig. 3.

PCE order | $N{}_{\mathit{train}}$ | $N{}_{\mathit{coeff}}$ |
---|---|---|

d = 3 | ||

p = 2 | 27 | 10 |

p = 3 | 64 | 20 |

p = 4 | 125 | 35 |

d = 5 | ||

p = 2 | 243 | 21 |

p = 3 | 1024 | 56 |

p = 4 | 3125 | 126 |

## 3. UQ analysis for beam transport lines

### 3.1 Variation in the quadrupole gradients

Parameter | Value |
---|---|

Order of surrogate model | p = 3 |

Number of inputs | d = 3 |

Input vector | $\mathit{\xi}=[{\mathrm{K}}_{\mathrm{QMD}10},{\mathrm{K}}_{\mathrm{QMD}11},{\mathrm{K}}_{\mathrm{QMD}12}]$ |

Number of coefficients | $\alpha $ = 20 |

Number of training points | 64 |

### 3.2 Uncertainties in the beam kinetic energy

Parameter | Reference | Tolerance |
---|---|---|

COMET energy (${\mathrm{E}}_{\mathrm{COMET}}$) | 250.0 MeV | $\pm $0.5 MeV |

COMET energy | 25 keV | $\pm $250 keV |

spread ($\mathrm{\Delta}$${\mathrm{E}}_{\mathrm{COMET}}$) | ||

Half-wedge thick. (${\mathrm{HW}}_{\mathrm{t}}$) | 2.5 mm | $\pm $0.2 mm |

## 4. UQ optimization in beam dynamics

### 4.1 Superconducting gantry project at PSI

COSY-Infinity.http://www.bt.pa.msu.edu/index_cosy.htm.

OPERA-3D.https://operafea.com/.

### 4.2 Beam envelope optimization with UQ

Parameter | Value |
---|---|

Order of surrogate model | p = 3 |

Number of inputs | d = 4 |

Input vector | $\mathit{\xi}=[{\text{SCQ}}_{y},\text{SCD}{2}_{x},\text{SCD}{2}_{y},\text{SCD}{2}_{B}]$ |

$\alpha $ = 35 | |

Number of training points | 256 |

Train. | Surrogate model | High-fidelity model | ||
---|---|---|---|---|

point ID | x-plane | y-plane | x-plane | y-plane |

11 | 2.38 | 2.41 | 1.72 | 4.88 |

13 | 2.67 | 2.57 | 2.01 | 3.35 |

## 5. Conclusions

## Acknowledgment

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COSY-Infinity.http://www.bt.pa.msu.edu/index_cosy.htm.

OPERA-3D.https://operafea.com/.