# Combining Wave and Particle Effects in the Simulation of X-ray Phase Contrast—A Review

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## Abstract

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## 1. Introduction

## 2. Wave Optics Simulation of X-ray Phase-Contrast Imaging

## 3. Monte Carlo Simulation of X-ray Phase-Contrast Imaging

## 4. Combined Wave-Optics and Monte Carlo Simulation of X-ray Phase Contrast

#### 4.1. Simulation of GB-XPCI by Combining Fresnel Propagation and MC Scattering

#### 4.2. Combining MC with Coherent Wave Optics via a Step-by-Step Simulation of GI XPCI

#### 4.3. Semi-Classical Monte Carlo Algorithm for the Simulation of X-ray Grating Interferometry

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## List of Acronyms

CNR | Contrast-to-noise ratio |

GBI | Grating-based interferometry |

GB-XPCI | Grating-based X-ray phase-contrast imaging |

GI | Grating interferometry |

MoBI | Modulation-based interferometry |

MC | Monte Carlo |

PBI | Propagation-based interferometry |

WDF | Wigner distribution function |

WBRT | Wigner-based ray tracing |

XPCI | X-ray phase-contrast imaging |

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**Figure 1.**The different steps of the simulation presented by Bartl. et al. [21] (Section 4.1) can be represented by a two-part simulation that is added at the detector: the coherent effects are simply added to the incoherent effects.

**Figure 2.**The different steps of the simulation presented by Peter et al. [22] (Section 4.2) are separated into their specific parts, depicting how the framework combines the particle and the wave effects within one simulation.

**Figure 3.**The different steps of the semi-classical MC algorithm presented by Tessarini et al. [23] (Section 4.3). It combines both wave and particle effects through the use of basic quantum mechanical concepts (indistinguishability of paths for a photon to arrive at the detector) and classical approximations.

Comparison of XPCI Simulation Methods | |||||
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Classical Methods | Pure MC Methods | Combination | |||

Benefits | Limitations | Benefits | Limitations | Benefits | Limitations |

Considers refraction and absorption | Formulations are not discrete | Photon by photon image construction possible | Interference and diffraction are not fully taken into account | Interference and scattering are possible in addition to refraction and absorption | |

Numerous simulations offer different implementations | Interference, diffraction and scattering are not taken into account | Scattering can be taken into account | Nonlinear effects and the complexities of modern optical systems can challenge accuracy | Two-part simulation can save time | |

Easy to implement: involves either the intensity of the Fourier transform of the wave or the intensity of a convolution of the wave with a propagator function | Images cannot be constructed photon by photon | Accurate simulation of phase-sensitive X-ray imaging | Photon-by-photon image construction possible | ||

Nonlinear effects and the complexities of modern optical systems can challenge accuracy | Assumption of independence | Accurate simulation of phase-sensitive X-ray imaging | |||

Convergence issues | Good agreement between measurements and simulations |

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**MDPI and ACS Style**

Pietersoone, E.; Létang, J.M.; Rit, S.; Brun, E.; Langer, M.
Combining Wave and Particle Effects in the Simulation of X-ray Phase Contrast—A Review. *Instruments* **2024**, *8*, 8.
https://doi.org/10.3390/instruments8010008

**AMA Style**

Pietersoone E, Létang JM, Rit S, Brun E, Langer M.
Combining Wave and Particle Effects in the Simulation of X-ray Phase Contrast—A Review. *Instruments*. 2024; 8(1):8.
https://doi.org/10.3390/instruments8010008

**Chicago/Turabian Style**

Pietersoone, Emilie, Jean Michel Létang, Simon Rit, Emmanuel Brun, and Max Langer.
2024. "Combining Wave and Particle Effects in the Simulation of X-ray Phase Contrast—A Review" *Instruments* 8, no. 1: 8.
https://doi.org/10.3390/instruments8010008