# Analysis of Numerical Artifacts Using Tetrahedral Meshes in Low Frequency Numerical Dosimetry
## 6 Conclusions  
Several exposure scenarios were analyzed in order to study whether tetrahedral meshes were capable of suppressing numerical errors caused by stair-casing approximation  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Max Value**} & \multicolumn{3}{c}{**99.9 th**} & \multicolumn{3}{c}{**99 th**} \\
**Tissue** & \multicolumn{3}{c}{**Voxel**} & \multicolumn{3}{c}{**Voxel**} & \multicolumn{3}{c}{**Voxel**} & \multicolumn{3}{c}{**Tetra**} & \multicolumn{3}{c}{**Voxel**} \\  & **1 mm** & **0.5 mm** & **Tetra** & **1 mm** & **0.5 mm** & **Tetra** & **1 mm** & **0.5 mm** & **Tetra** \\ \hline 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 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Deviation of the \(E\)-field in the localized case—all values are expressed in mV/m.  
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**).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Max Value**} & \multicolumn{3}{c}{**99.9 th**} & \multicolumn{3}{c}{**99 th**} \\
**Tissue** & \multicolumn{3}{c}{**Voxel**} & \multicolumn{3}{c}{**Voxel**} & \multicolumn{3}{c}{**Voxel**} & \multicolumn{3}{c}{**Tetra**} & \multicolumn{3}{c}{**Voxel**} \\  & **1 mm** & **0.5 mm** & **Tetra** & **1 mm** & **0.5 mm** & **Tetra** & **1 mm** & **0.5 mm** & **Tetra** \\ \hline 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 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Deviation of the \(E\)-field in the homogeneous case—all values are expressed in mV/m.  
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.  
errors in voxelized models when curved boundaries are approximated with voxels. From the analysis, we discovered that both voxelized and tetrahedral head models suffered from artifacts in the evaluation of high electric field values. In [15], a similar conclusion was obtained but the cause of the artifacts was mainly associated with the presence of low quality tetrahedra.  
In this paper, we showed that a tetrahedral mesh is indeed able to remove the source of computational artifacts related to the geometrical modeling of the computational domain (stair-casing), however, in a real exposure scenario, other sources of numerical artifacts are still present. These numerical artifacts are related to two fundamental factors: tetrahedral mesh quality and tissue contrast effect. It was shown that the conductivity contrast between neighboring tissues can cause very high electric fields even in tetrahedra with good quality index. For this reason, even when the mesh quality is close to ideal, it is almost impossible to avoid the crossing of induced currents across tissue interfaces in real exposure scenarios. As a consequence, the use of filtering techniques cannot be completely avoided.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.