The paper presents the numerical results for the induced electric field in the various models of the human eye and the head. The comparison between the extracted or the single organ models and the compound organ models placed inside realistic head models obtained from the magnetic resonance imaging scans is presented. The numerical results for several frequencies and polarizations of the incident electromagnetic (EM) plane wave are obtained using the hybrid finite element method/boundary element method (FEM/BEM) formulation and the surface integral equation (SIE) based formulation featuring the use of method of moments, respectively. Although some previous analysis showed the similar distribution of the induced electric field along the pupillary axis obtained in both eye models, this study showed this not to be the case in general. The analysis showed that the compound eye model is much more suitable when taking into account the polarization of the incident EM wave. The numerical results for the brain models showed much better agreement in the maximum values and distributions of the induced surface field between detailed models, while homogeneous brain model showed better agreement with the compound model in the distribution along selected sagittal axis points. The analysis could provide some helpful insights when carrying out the dosimetric analysis of the human eye and the head/brain exposed to high frequency EM radiation.
The electromagnetic (EM) fields generated by the various wireless communication equipment such as mobile phones and base station antennas have increased the concern among the general population related to the possible harmful effects. As the established biological effect of high frequency (HF) electromagnetic fields is tissue heating, the assessment of this HF exposure is based on determining the specific absorption rate (SAR) that is related to the electric field induced in the tissue. The HF exposure assessment is particularly important in the case of human eye and brain since experimental measurement in healthy humans is very difficult if not impossible. The solution to this is the use of the computational models and the related numerical solutions as a tool for assessment of HF exposure [
The computational models employed for this particular type of assessment can be classified as realistic models of the human body (or particular organs of interest) based on the magnetic resonance imaging (MRI) (e.g. [
The detailed models of the complete human body are nowadays readily available (e.g., [
The selection between the simplified single organ model and the more detailed and complete body model is not simple nor straightforward. This paper is an extension of recent study [
The paper is organized as follows. The first part gives a description of the extracted and the compound eye and brain models, respectively. Following this, the description of the hybrid finite element method/boundary element method (FEM/BEM) approach [
The human eye is a delicate organ consisting of many fine parts, each performing various important functions. In order to account for such a small but important subtleties, hence, a detailed model of the eye is very important. The sagittal cross-section of the human eye depicting its parts is shown in Figure
Sagittal cross-section of the human eye depicting various tissues.
Magnetic resonance imaging (MRI) can very accurately capture details of the human anatomy; however, the spatial resolution of MRI is not sufficient to capture the fine geometrical details of the human eye. According to some recent studies, ultra-high resolution 7 T MRI techniques were able to capture spatial anatomical data with isotropic resolutions of 0.6 mm and 0.7 mm [
Therefore, the detailed geometrical model of the human eye, named here as the extracted (single) eye model, has been developed from the available MRI scans as well as from various medical measurements data. Modeled tissues from the extracted and the compound eye models are shown in Figure
Modeled tissues from the extracted and the compound eye models.
The extracted eye model consists of 16 tissues, whose parameters are given in Table
Tissue parameters used in eye and head models.
Tissue | 900 MHz | 1800 MHz |
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Brainstem | 0.622 | 38.577 | 0.915 | 37.011 | 1043 |
Cerebellum | 1.308 | 48.858 | 1.709 | 46.114 | 1039 |
Head skin | 0.899 | 40.936 | 1.185 | 38.872 | 1050 |
Liquor | 1.667 | 68.875 | 2.032 | 68.573 | 1035 |
Skull | 0.364 | 20.584 | 0.588 | 19.343 | 1900 |
Mandible | 0.364 | 20.584 | 0.588 | 19.343 | 1900 |
Grey matter | 0.985 | 52.282 | 1.391 | 50.079 | 1039 |
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Anterior chamber | 1.667 | 68.875 | 2.032 | 68.573 | 1003 |
Choroid | 0.729 | 44.561 | 1.066 | 43.343 | 1000 |
Ciliary body | 0.978 | 54.811 | 1.341 | 53.549 | 1040 |
Cornea | 1.438 | 54.835 | 1.858 | 52.768 | 1076 |
Iris | 0.978 | 54.811 | 1.341 | 53.549 | 1040 |
Ligaments | 0.760 | 45.634 | 1.201 | 44.252 | 1000 |
Ora serrata | 0.882 | 45.711 | 1.232 | 43.850 | 1000 |
Posterior chamber | 1.667 | 68.875 | 2.032 | 68.573 | 1000 |
Retina | 1.206 | 55.017 | 1.602 | 53.568 | 1039 |
Sclera | 1.206 | 55.017 | 1.602 | 53.568 | 1076 |
Vitreous body | 1.667 | 68.875 | 2.032 | 68.573 | 1009 |
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Lens-I | 0.824 | 46.399 | 1.147 | 45.353 | 1100 |
Lens-II | 0.824 | 47.011 | 1.147 | 45.925 | 1100 |
Lens-III | 0.824 | 47.694 | 1.147 | 46.221 | 1100 |
Lens-IV | 0.824 | 48.383 | 1.147 | 46.883 | 1100 |
Lens-V | 0.824 | 49.076 | 1.147 | 47.554 | 1100 |
The crystalline lens is modeled using five layers with varying relative permittivity, according to the Gradient Refraction Index (GRIN) model [
The boundary surface of the eye model is discretized using 7.986 triangular elements, while interior domain of the eye is discretized using 415.429 tetrahedral elements.
The extracted eye model developed in the previous section is incorporated in the full model of the human head composed of various head tissues as shown in Figure
Models for the brain comparison: (a) homogeneous brain model, (b) three-compartment head model, and (c) compound model. Overlay on two latter models is showing various head tissues surrounding the brain.
The detailed model of the human head was constructed from the MRI of a 24-year-old male [
As a very simple representation of the human brain, shown in Figure
Frequency dependent parameters of the homogeneous models are taken from [
It should be noted that the employed brain geometry is an extremely simplified model, as the brain surface was radically smoothed thus missing complex folding structures of gyri and sulci, in addition to consisting of a single, homogeneous structure. Nonetheless, it is important to emphasize that this homogeneous model could still be useful as in the initial assessment comparison [
Although it is more realistic than the sphere, it lacks the detailed cortical structures and inhomogeneity (grey/white matter, ventricles, etc.). In order to overcome this limitation, the future work should therefore include comparisons on detailed anatomically correct head model, featuring complex material maps and shapes. Nonetheless, regarding the use of the homogeneous model it is important to emphasize that it is reasonable to start comparing different numerical techniques using simple models thus opening the subject.
The next logical step in the development of a more realistic brain model would be to place it inside the various surrounding tissues. The most realistic head model routinely used in experimental magnetoencephalography (MEG) is the so-called three-shell or the three-compartment model [
Tissue parameters used in the three-compartment model.
Tissue | 900 MHz | 1800 MHz |
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Scalp | 0.899 | 40.936 | 1.185 | 38.872 | 1050 |
Brain | 0.985 | 52.282 | 1.391 | 50.079 | 1039 |
Skull | 0.364 | 20.584 | 0.588 | 19.343 | 1900 |
Electromagnetic wave incident on the human eye or head can be treated as unbounded scattering problem. Using the Stratton-Chu integral expression, the time harmonic electric field in the domain exterior to the head is expressed by the following boundary integral equation [
Performing some manipulations, (
The form of (
The fields
The unknown coefficients
Vector base function
After the weighted residual approach is applied to (
After applying some standard vector identities, followed by the divergence theorem, the weak form is obtained:
The FEM/BEM coupling can now be employed by using the continuity of the tangential components of electric and magnetic fields across the surface
After substituting
Inserting (
After having discretized the interior domain into finite set of tetrahedrons, and using the expansion given by (
The element matrix
Finally, the element matrix
The procedure for calculating the surface integrals from (
The human brain exposure can be estimated by means of coupled surface integral equations (SIE) [
Set of integral equations (
Applying the weighted residual approach, that is, multiplying (
The first set of numerical results is obtained using the hybrid FEM/BEM formulation. The electric field induced in the extracted and the compound model of the eye, respectively, are given in Figures
Induced electric field due to 1 GHz EM wave on the surface of the eye (anterior and top view). (a, c) Compound eye model: (a) horizontal polarization and (c) vertical polarization. (b, d) Extracted eye model: (b) horizontal polarization and (d) vertical polarization.
Induced electric field due to 1800 MHz EM wave on the surface of the eye (anterior and top view). (a, c) Compound eye model: (a) horizontal polarization and (c) vertical polarization. (b, d) Extracted eye model: (b) horizontal polarization and (d) vertical polarization.
Induced electric field in the transverse cross-section of the compound eye model (a, c) and the extracted eye model (b, d). Incident EM wave of (a) and (b) 1 GHz horizontal (left) and vertical (right) polarization and (c) and (d) 1800 MHz horizontal (left) and vertical (right) polarization.
Comparison of the induced electric field along the pupillary axis of the compound and the extracted eye models, respectively, (a) 1 GHz, both polarizations; (b) 1800 MHz, both polarizations.
Figure
The similar finding was shown for the 1800 MHz case, as shown in Figure
It should also be noted that the extracted eye model gave highest values in the posterior parts denoting sclera, while the compound model obtained similar trend at both frequencies; that is, the highest obtained values are in the cornea.
More details on the distribution of the induced field inside the eye models can be seen in Figures
The results from the previous study [
One interesting fact that can be seen from Figure
The following set of numerical results are obtained using the homogeneous brain model solved using the SIE/MoM formulation, while the results for the three-compartment model and the compound brain/head model are obtained using the hybrid FEM/BEM formulation. The total of four different situations is considered, that is, that of both vertically and horizontally polarized plane wave of 900 MHz and 1800 MHz and amplitude of 1 V/m. The incident EM wave is directed toward the anterior part of the brain/head models.
The induced electric field on the brain surfaces of the three models is given in Figures
Induced electric field on the surface of the brain due to 900 MHz horizontally polarized EM wave: (a) homogeneous model, (b) three-compartment model, and (c) compound model.
Induced electric field on the surface of the brain due to 900 MHz vertically polarized EM wave: (a) homogeneous model, (b) three-compartment model, and (c) compound model.
Induced electric field on the surface of the brain due to 1800 MHz horizontally polarized EM wave: (a) homogeneous model, (b) three-compartment model, and (c) compound model.
Induced electric field on the surface of the brain due to 900 MHz vertically polarized EM wave: (a) homogeneous model, (b) three-compartment model, and (c) compound model.
One drawback of the current implementation of the homogeneous model is a relatively low number of triangular elements used for the discretization of the brain surface; that is, the brain surface is tessellated using the
Comparison of the induced electric field along the sagittal axis of the homogeneous, the three-compartment, and the compound models, respectively, due to 900 MHz EM wave, horizontal polarization, and incident on the anterior side.
Comparison of the induced electric field along the sagittal axis of the homogeneous, the three-compartment, and the compound models, respectively, due to 900 MHz EM wave, vertical polarization, and incident on the anterior side.
Comparison of the induced electric field along the sagittal axis of the homogeneous, the three-compartment, and the compound models, respectively, due to 1800 MHz EM wave, horizontal polarization, and incident on the anterior side.
Comparison of the induced electric field along the sagittal axis of the homogeneous, the three-compartment, and the compound models, respectively, due to 1800 MHz EM wave, vertical polarization, and incident on the anterior side.
On the other hand, the results obtained using the two more elaborate models showed similar distributions for induced surface fields, as evident in Figures
More details on the distribution of the electric field along the sagittal axis of the three brain models, obtained approximately at the medial prefrontal cortex, are seen in Figures
The results from both 1800 MHz cases showed that the three-compartment model obtained very uniform distribution of the field in the central part of the brain. In contrast to this, the homogeneous model showed field distribution in the central part of the brain much similar to the detailed compartment model.
The paper presented the comparison of the induced electric field in the extracted or the single organ models of the human eye and the brain, and the compound organ models incorporated in the detailed human head, respectively. The models of the human eye and brain were exposed to high frequency electromagnetic radiation of several frequencies and polarizations. The numerical results obtained using the hybrid FEM/BEM formulation showed similar distributions of the induced electric field along the pupillary axis in both the extracted and the compound eye models only in the case of 1 GHz vertically polarized wave. However, the results for horizontal polarization and also for 1800 MHz showed this not to be the case. Additionally, the study showed that extracted eye model due to its symmetrical nature obtained the similar results for both polarizations. The compound eye model showed different results, suggesting to be more appropriate when taking into account polarization of the incident electromagnetic wave.
The numerical results for the brain models showed similar maximum values and surface distributions for the three-compartment model and the compound model, while homogeneous model obtained significantly lower values. On the other hand, the homogeneous model obtained field distributions along the brain sagittal axis much similar to the compound model, while the three-compartment model at 1800 MHz obtained somewhat unexpectedly a very uniform distribution in the brain tissue, dissimilar to the other two models.
The authors declare that they have no conflicts of interest.
The authors would like to thank I. Laakso from Aalto University, Finland, and A. Hirata from Nagoya Institute of Technology, Japan, for providing the detailed head model, and also A. Hunold and J. Haueisen from TU Ilmenau, Germany, for providing the three-compartment head model.