# Analysis of Numerical Artifacts Using Tetrahedral Meshes in Low Frequency Numerical Dosimetry

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## Abstract

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## 1. Introduction

^{®}(VHP), and the Ella phantom based on the Virtual Family, whose tetrahedral mesh was generated from the $1\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ voxel model by means of the free Matlab toolbox iso2mesh [12]. It is interesting to note that in these papers the authors’ attention is focused on the validation of the new proposed formulations rather than the effects of tetrahedral meshes on numerical artifacts. The question of the influence of the quality of the mesh on computational results is explicitly addressed in [13], where it is proposed to use a local a posteriori residual estimator to evaluate the error.

## 2. Numerical Formulation

## 3. Tetrahedral Mesh Quality

## 4. Dosimetric Models

#### 4.1. Multilayered Sphere

#### 4.2. Human Head—Anatomical 3D Model

#### 4.3. Human Head—Simplified 3D Models

## 5. Numerical Results

#### 5.1. Multilayered Sphere

#### 5.2. Human Head—Simplified 3D Model

#### 5.3. Human Head—Complete 3D Model

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**No stair-casing effect on curved boundaries thanks to tetrahedral discretization (

**a**). Stair-casing effect on curved boundaries due to the use of a voxel-based mesh (

**b**).

**Figure 2.**Tetrahedron with the inradius r and the circumradius R, respectively, drawn in orange and blue.

**Figure 3.**Multilayered sphere. The various layers are highlighted by different colors. In the inset the stair-casing effect due to the voxel-discretization is displayed. The source field is created by a coil with a $5\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}$ radius and located $13\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}$ above the center of sphere.

**Figure 4.**Colin27 Average Brain: 3D model (

**a**), 2D cut-plane (

**b**), axisymmetric model (

**c**) and planar model (

**d**).

**Figure 5.**Manual control of the tetrahedral mesh: the boundaries were smoothed to avoid singularities. Tissue boundaries before local smooth edits (

**a**) and the related mesh (

**b**). Tissue boundaries after local smooth edits (

**c**) and the related mesh (

**d**).

**Figure 6.**Localized exposure (coil in front of the head model). The induced electric field in Colin27 Average Brain is shown for tetrahedral mesh and voxel-based meshes with resolution of $1\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ e $0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ in each tissue: skull (

**a**), CSF (

**b**), grey matter (

**c**) and white matter (

**d**).

**Figure 7.**Tetrahedral mesh quality of Colin27 3D model. The quality index is on the x-axis, while the number of tetrahedral elements corresponding to the respective mesh quality index is shown on the y-axis.

**Figure 8.**Tetrahedron with $q=0.9$ related to the maximum electric field in CSF (

**a**). Tetrahedron with $q=0.11$ related to the maximum electric field in the white matter (

**b**).

Layer | Tissue | Radius (mm) | Conductivity (S/m) |
---|---|---|---|

1 | Skin | 80 | $0.0002$ |

2 | Fat | 76 | $0.043$ |

3 | Muscle | 74 | $0.34$ |

4 | Skull | 72 | $0.02$ |

5 | Muscle | 68 | $0.34$ |

6 | Cerebrospinal fluid | 66 | $2.0$ |

7 | Brain | 64 | $0.11$ |

8 | Cerebrospinal fluid | 42 | $2.0$ |

9 | Brain | 38 | $0.11$ |

**Table 2.**Deviation between analytical and computed induced electric field on tetrahedral and voxel-based mesh.

Tissue | Analytical | Voxel 1 mm | Tetra | ||
---|---|---|---|---|---|

Solution | Max | 99.9th | 99th | Max | |

(mV/m) | $({E}/{{E}}_{\mathbf{analytical}})$ | $({E}/{{E}}_{\mathbf{analytical}})$ | |||

Skin | 16.38 | 1.1563 | 1.0534 | 0.9844 | 1.0051 |

Fat | 14.37 | 1.1104 | 1.0804 | 1.0265 | 1.0050 |

Muscle | 13.47 | 1.1344 | 1.0618 | 1.0114 | 1.0034 |

Skull | 12.63 | 1.2132 | 1.1223 | 1.0192 | 1.0026 |

Muscle | 11.11 | 1.1524 | 1.0910 | 1.0294 | 1.0041 |

CSF | 10.40 | 1.1836 | 1.0916 | 1.0187 | 1.0051 |

Brain | 9.78 | 1.2699 | 1.0971 | 0.9373 | 1.0022 |

CSF | 4.76 | 1.1721 | 1.1001 | 1.0016 | 0.9988 |

Brain | 4.12 | 1.2389 | 1.1284 | 0.9878 | 1.0015 |

Tissue | Reference | Voxel 1 mm | Voxel 0.5 mm | Tetra | ||||
---|---|---|---|---|---|---|---|---|

2D | Max | 99.9th | 99th | Max | 99.9th | 99th | Max | |

(mV/m) | $({E}/{{E}}_{2}{D})$ | $({E}/{{E}}_{2}{D})$ | $({E}/{{E}}_{2}{D})$ | |||||

Skull | 2.79 | 1.372 | 1.149 | 1.018 | 1.263 | 1.111 | 0.994 | 1.002 |

CSF | 2.31 | 1.484 | 1.208 | 1.100 | 1.283 | 1.152 | 1.062 | 1.003 |

Grey Matter | 2.26 | 1.509 | 1.279 | 1.110 | 1.438 | 1.225 | 1.050 | 1.001 |

White Matter | 2.12 | 1.025 | 1.002 | 0.975 | 1.010 | 0.998 | 0.975 | 1.003 |

Tissue | Reference | Voxel 1 mm | Voxel 0.5 mm | Tetra | ||||
---|---|---|---|---|---|---|---|---|

2D | Max | 99.9th | 99th | Max | 99.9th | 99th | Max | |

(mV/m) | $({E}/{{E}}_{2}{D})$ | $({E}/{{E}}_{2}{D})$ | $({E}/{{E}}_{2}{D})$ | |||||

Skull | 6.89 | 1.086 | 0.854 | 0.596 | 1.053 | 0.738 | 0.564 | 1.092 |

CSF | 6.19 | 1.004 | 0.986 | 0.623 | 1.039 | 0.866 | 0.671 | 1.098 |

Grey Matter | 11.06 | 0.805 | 0.694 | 0.500 | 0.769 | 0.610 | 0.452 | 0.934 |

White Matter | 4.68 | 1.045 | 0.978 | 0.894 | 1.018 | 0.925 | 0.843 | 1.006 |

Tissue | Max Value | 99.9th | 99th | ||||||
---|---|---|---|---|---|---|---|---|---|

Voxel | Tetra | Voxel | Tetra | Voxel | Tetra | ||||

1 mm | 0.5 mm | 1 mm | 0.5 mm | 1 mm | 0.5 mm | ||||

Skull | 12.12 | 10.49 | 6.59 | 4.21 | 4.17 | 3.72 | 3.51 | 3.47 | 3.29 |

CSF | 6.07 | 6.92 | 4.91 | 3.94 | 3.74 | 3.72 | 3.19 | 3.17 | 3.25 |

GM | 12.09 | 14.11 | 8.12 | 4.97 | 5.05 | 5.04 | 3.69 | 3.63 | 3.72 |

WM | 6.58 | 9.86 | 4.83 | 3.40 | 3.31 | 3.39 | 2.84 | 2.79 | 2.90 |

Tissue | Max Value | 99.9th | 99th | ||||||
---|---|---|---|---|---|---|---|---|---|

Voxel | Tetra | Voxel | Tetra | Voxel | Tetra | ||||

1 mm | 0.5 mm | 1 mm | 0.5 mm | 1 mm | 0.5 mm | ||||

Skull | 16.95 | 17.23 | 10.45 | 8.30 | 8.29 | 6.93 | 6.02 | 6.01 | 5.44 |

CSF | 9.30 | 11.97 | 12.38 | 5.99 | 5.85 | 5.88 | 4.87 | 4.85 | 5.10 |

GM | 17.05 | 23.94 | 17.70 | 8.07 | 8.17 | 9.10 | 5.91 | 5.82 | 5.94 |

WM | 11.63 | 17.10 | 30.14 | 5.67 | 5.55 | 5.62 | 4.72 | 4.64 | 4.81 |

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**MDPI and ACS Style**

Conchin Gubernati, A.; Freschi, F.; Giaccone, L.; Scorretti, R.
Analysis of Numerical Artifacts Using Tetrahedral Meshes in Low Frequency Numerical Dosimetry. *Appl. Sci.* **2022**, *12*, 6526.
https://doi.org/10.3390/app12136526

**AMA Style**

Conchin Gubernati A, Freschi F, Giaccone L, Scorretti R.
Analysis of Numerical Artifacts Using Tetrahedral Meshes in Low Frequency Numerical Dosimetry. *Applied Sciences*. 2022; 12(13):6526.
https://doi.org/10.3390/app12136526

**Chicago/Turabian Style**

Conchin Gubernati, Alice, Fabio Freschi, Luca Giaccone, and Riccardo Scorretti.
2022. "Analysis of Numerical Artifacts Using Tetrahedral Meshes in Low Frequency Numerical Dosimetry" *Applied Sciences* 12, no. 13: 6526.
https://doi.org/10.3390/app12136526