# Analysis of Numerical Artifacts Using Tetrahedral Meshes in Low Frequency Numerical Dosimetry
## 5 Numerical Results
### Human Head--Complete 3D Model  
In this section, exposure of a realistic head model is considered in two scenarios: (1) homogeneous magnetic flux density with a single component in the \(z\)-direction of 200 uT, (2) magnetic flux density created by a coil with a radius of 20 cm. The coil lays in the \(xy\) plane and the distance from the head is 30 cm. The center of the coil is aligned with the head center and the current flowing in the coil is 1 kA. The induced electric field is computed using tetrahedral and voxel elements. In the tetrahedral mesh is the original available in [32] without additional refinement, containing about 70,200 nodes and 423,400 tetrahedra; in the voxel mesh there are about 4,140,000 nodes and 4,045,000 elements when the resolution is 1 mm, and about 32,753,000 nodes and 32,374,000 elements when the resolution is 0.5 mm. Results are provided in terms of maximum value, 99.9 th percentile and 99 th percentile for both voxel and tetrahedra in Table 5 (homogeneous case) and in Table 6. The induced electric field exhibits higher values in the skull and gray matter. A graphical comparison for the case of localized exposure scenario in several tissues is shown in Figure 6. The maps of the electric field for all meshes are in good agreement. In the case of homogeneous exposure, Table 5, it is possible to observe that the maximum value obtained with the tetrahedral mesh is always lower than the one obtained with voxels. However, when moving to the more realistic localized exposure this result is not confirmed for all tissues. In the cerebrospinal fluid (CSF) and white matter (WM) the tetrahedral elements lead to the highest maximum value of induced electric field. This means that, when staircasing is not the only source of numerical artifacts, tetrahedral meshes are also subject to numerical artifacts. To better understand the causes of the peak values obtained within the tetrahedral mesh, a correlation with the quality index \(q\) is carried out. First of all, the overall mesh quality is shown in Figure 7. It is possible to observe that most of the tetrahedral elements have an acceptable quality index (\(q>0.7\)). However, elements with a low quality index exist. The quality index \(q\) related to the tetrahedron with the maximum electric field value is equal to 0.9 in the cerebrospinal fluid (see Figure 8a), while it is equal to 0.11 in the white matter (see Figure 8b). This means that, while in the white matter the \"culprit\" for the numerical artifact is likely to be the irregular shape of the tetrahedron, the same conclusion does not apply to the CSF tissue. In fact, the maximum value in the CSF is related to a tetrahedron belonging to the boundary characterized by a high conductivity contrast (2 S/m in the CSF, 0.2 S/m in the neighbor tissues). In this case, the induced currents cross tissue boundaries causes numerical artifacts due to the large difference in \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & **Reference** & \multicolumn{4}{c}{**Voxel 1 mm**} & \multicolumn{4}{c}{**Voxel 0.5 mm**} & \multicolumn{2}{c}{**Tetra**} \\
**Tissue** & **2D** & **Max** & **99.9 th** & **99 th** & **Max** & **99.9 th** & **99 th** & **Max** \\  & **(mV/m)** & & **(E/E\({}_{\text{2D}}\))** & & & **(E/E\({}_{\text{2D}}\))** & & **(E/E\({}_{\text{2D}}\))** \\ \hline 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 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Deviation of the \(E\)-field in the planar case.