# Analysis of Numerical Artifacts Using Tetrahedral Meshes in Low Frequency Numerical Dosimetry
## 1 Introduction  
Low-frequency (LF) numerical dosimetry is an active research field due to some unresolved issues that still require to be addressed [1; 2]. The evaluation of human exposure to the electromagnetic field requires an evaluation of the maximum value of the induced electric field in the human body. The use of the raw solution is, however, subject to numerical artifacts stemming from different sources [3]. One common source of numerical artifacts is the so called stair-casing effect, due to the reconstruction of the body tissues using cubic elements (voxels) coming directly from DICOM (Digital Imaging and Communications in Medicine) images [4]. Voxel-based meshes are intrinsically unable to match curved boundaries, as shown in Figure 1.  
Numerical artifacts cause an overestimation of the exposure and must therefore be filtered out with specific methods proposed in standards, guidelines and the literature [5; 6; 7]. One possible solution to remove the stair-casing errors is the use of tetrahedral meshes. Over the last decades, thanks to the use of geometric modelling software, the organs and tissues of human models derived from DICOM images have been reconstructed using three-dimensional mathematical primitives (e.g., NURBS), giving the possibility to generate tetrahedral meshes that better reproduce the boundaries of curved surfaces. Human models discretized with tetrahedral meshes started to be used both in low-frequency and high-frequency numerical dosimetry (see, for example, [8] where a procedure to optimize the specific absorption rate proposed in a patient during oncological hyperthermia treatment is presented). In [9; 10], an anatomical model of the human body composed of tetrahedral elements and obtained from CT scans was used. The aim was to validate new methodologies based on A-\(\varphi\) finite elements formulation to compute the induced currents into thehuman body due to LF magnetic fields generated, respectively, by realistic devices and completely unknown power sources. In [11], two dual finite elements formulations to perform numerical dosimetry are presented. Two human models discretized with tetrahedral elements are used in the analysis of numerical errors related to the proposed methodology: the ZOL phantom built using the software AMIRA starting from the segmented data of the Visible Human Project(r) (VHP), and the Ella phantom based on the Virtual Family, whose tetrahedral mesh was generated from the 1 mm voxel model by means of the free Matlab toolbox iso2 mesh [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. A comparison between tetrahedral and voxel-based meshes is the main objective of this paper. Regarding this aspect, some results are already available in the literature. In [14], the authors validated their method on the tetrahedral human model ZOL and the numerical results are compared with some existing data evaluated on voxelized human models. The comparison showed some inconsistencies between the data obtained with the tetrahedral models and with the voxel-based ones. The authors underlined the difficulties in comparing different models and methods due to the fact that discretization and post-processing techniques play an important role in the process. However, they did not perform a detailed analysis on the nature of the numerical artifacts present in tetrahedral and voxel-based meshes and no conclusions can be drawn from their analysis. In [15], the authors compared the use of voxel and tetrahedral meshes with the idea of eliminating the existing effect with tetrahedral meshes. Computations were performed using five 3D head models obtained from magnetic resonance imaging.