# A Study on the Use of Compound and Extracted Models in the High Frequency Electromagnetic Exposure Assessment

Authors: Mario Cvetković; Hrvoje Dodig; Dragan Poljak
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1–3].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. [4]) or the simplified models, computationally much less demanding but failing to provide accurate results in most of the exposure scenarios [5].The detailed models of the complete human body are nowadays readily available (e.g., [6, 7]); however, the detailed human body model puts the significant burden in the computational model preparation at the same time putting strain on the available computational resources. In addition to this, there are cases when only the particular organ or body parts are of research interest, as when the initial assessment is considered.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 [8] on the comparison of the results between the compound and the extracted eye models, respectively, exposed to 1 GHz electromagnetic plane wave at single polarization. This paper deals with high frequency EM exposure in case of the human eye and human head/brain exposed to several frequencies and polarizations. The single organ models of the eye and the brain, termed here the extracted or the single models, and those models incorporated in the detailed human head model, termed the compound models, are used in the numerical assessment of the induced electric field due to high frequency electromagnetic radiation.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 [9] and the surface integral equation (SIE) approach [10], respectively, are given. The corresponding numerical solutions are reported as well. The numerical results for the induced electric field in two eye models exposed to 1 GHz and 1800 MHz EM plane wave are given in the following section. The same section features another set of results related to the induced field in three human brain/head models exposed to 900 MHz and 1800 MHz EM plane wave. The discussion of the results and the conclusion are given in the final part.


## 2. Materials and Methods

### 2.1. Extracted (Single) Eye Model

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 Figure1.Figure 1
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 [11] while [12] reported spatial resolutions of 0.5×0.5×0.6 mm3. To the best of the authors knowledge, the most accurate MRI scans were performed using the 9.4 T magnet [13] achieving voxel volumes of about 0.13×0.13×0.8 mm3. However, in order to capture the fine details of the eye tissues such as choroid, retina, and iris, a much finer resolution is required. It is estimated that in order to have at least several layers of MRI voxels in the retina, one would need spatial MRI resolution of about 30 × 30 × 30 μm3 or less, or the resulting boundary surfaces between tissues would have become staircased.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 Figure2.Figure 2
Modeled tissues from the extracted and the compound eye models.The extracted eye model consists of 16 tissues, whose parameters are given in Table1. The frequency dependent dielectric parameters (the electrical conductivity σ and the relative permittivity ε) are modeled using the 4-Cole-Cole method [14]. Table 1 gives also the tissue mass density ρ.Table 1
Tissue parameters used in eye and head models.
Tissue
900 MHz
1800 MHz
ρ (g/m3)
σ (S/m)
ε (-)
σ (S/m)
ε (-)
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
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
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
1100The crystalline lens is modeled using five layers with varying relative permittivity, according to the Gradient Refraction Index (GRIN) model [15, 16].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.


### 2.2. Compound Eye and Head Model

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 Figure3(c). This model is named the compound eye model.Figure 3
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.
(a)
(b)
(c)The detailed model of the human head was constructed from the MRI of a 24-year-old male [17]. The current implementation of the model features 7 tissues in addition to 16 ocular tissues from the extracted eye model. The head tissue parameters are also modeled using the 4-Cole-Cole method [14] and are given in Table 1. The boundary surface of the complete head model is discretized using 5.934 triangular elements, while region inside the head is discretized using 4.073.250 tetrahedral elements.


### 2.3. Homogeneous Brain Model

As a very simple representation of the human brain, shown in Figure3(a), the homogenized realistic-shaped brain reported in [10] was used. To facilitate the preparation of otherwise a very complex organ, the freely available model from Google SketchUp was scaled to adult average brain using dimensions of length 167 mm, width 140 mm, height 93 mm, and volume of 1400 cm3 [18]. The surface of the model was discretized using the triangular elements, enabling the use of Rao-Wilton-Glisson (RWG) basis functions specially developed for triangular patches.Frequency dependent parameters of the homogeneous models are taken from [10]. In addition, linear and isotropic behavior is assumed for the electrical properties of tissues.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 [19].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.


### 2.4. Three-Compartment Head Model

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 [20], consisting of homogeneous compartments of brain, skull, and scalp, as depicted in Figure 3(b). This model is freely available and can be easily implemented in the BEM analysis [20]. The parameters of the three-shell model are given in Table 2.Table 2
Tissue parameters used in the three-compartment model.
Tissue
900 MHz
1800 MHz
ρ (g/m3)
σ (S/m)
ε (-)
σ (S/m)
ε (-)
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


### 2.5. Hybrid FEM/BEM Method

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 [9, 21]:(1)αE→ext′=E→inc′+∮∂Vn^×∇×E→extGdS+∮∂Vn^×E→ext×∇G+n^·E→ext∇GdS,where n^ is an outer normal to surface ∂V bounding the volume V and α is the solid angle subtended at the observation point. The total and the incident electric field are denoted by E→ext and E→inc, respectively, and G is the free space Green’s function given by(2)G=Gr→,r→′=e-jkR4πR;R=r→-r→′,where R is the distance from the observation point r→ to the source point r→′ and k denotes the wave number.Performing some manipulations, (1) can be specified in terms of tangential components of electric field E→ and magnetic field H→ on the boundary surface ∂V [9]:(3)αE→ext′=E→inc′-jωμ∮∂Vn^×H→extGdS+∮∂Vn^×E→ext×∇G-1σ+jωμ∇s·n^×H→ext∇GdS.The form of (3) is pertinent for coupling to the governing differential equations of the interior inhomogeneous domain, given by [9]:(4)∇×jωμ∇×E→int-σ+jωεE→int=0.The fieldsE→ and H→ are approximated using the edge elements [22] preserving the tangential continuity of the fields on the boundary(5)E→=∑i=1nδiw→iei,H→=∑i=1nδiw→ihi.The unknown coefficientsei and hi, respectively, associated with each edge of the model, are determined from the global system of equations while the coefficient δi=±1 is equal to 1 depending on whether the direction of local edge coincides with the chosen global edge direction.Vector base functionwk is given by [23] w→k=Ni∇Nj-Nj∇Ni, where Ni and Nj are the first order barycentric shape functions.After the weighted residual approach is applied to (4), followed by the dot product of (4) with test function wi, and using the Galerkin-Bubnov procedure, it can be written that(6)∫Vδiw→i·∇×jωμ∇×E→int-σ+jωεE→intdV=0.After applying some standard vector identities, followed by the divergence theorem, the weak form is obtained:(7)∫Vjωμ∇×δiw→i·E→int-σ+jωεδiw→i·E→intdV=∮∂VdS→·δiw→i×H→int.The FEM/BEM coupling can now be employed by using the continuity of the tangential components of electric and magnetic fields across the surface∂V. This leads to n^×E→int=n^×E→ext and n^×H→int=n^×H→ext. Thus, it can be written that(8)∮∂VdS→′·δiw→i×E→int′=∮∂VdS→′·δiw→i×E→ext′.After substitutingE→ext and H→ext in (3) with E→int and H→int, respectively, and after inserting (3) into (8), the following double surface integral is obtained:(9)∮∂VdS→′·δiw→i×αE→ext′=∮∂VdS→′·δiw→i×E→inc′-jωμ∮∂VdS→′·δiw→i×∮∂Vn^×H→intGdS+∮∂VdS→′·δiw→i×∮∂Vn^×E→int×∇GdS-1ω+jωμ∮∂VdS→′·δiw→i×∮∂V∇s·n^×H→ext∇GdS.Inserting (5) into (9) results in the system of equations related to the edges at the boundary surface of the problem:(10)Ebemebem=einc+HbemhbemEfemefem=Hfemhfem,where ebem and hbem are unknown coefficients associated with boundary surface of the scattering problem, einc are known coefficients calculated from the incident field, and matrices [Ebem] and [Hbem] arise from boundary integral equation (9), while matrices [Efem] and [Hfem] stem from FEM (7).After having discretized the interior domain into finite set of tetrahedrons, and using the expansion given by (5), followed by the Galerkin procedure, the finite element matrices could be written. Thus, the element matrix Ke of the first term from (7), after some work, can be written as(11)1kB∑i=16∑j=16∇×wi→δi·∇×wj→δjaidV=VΔkBk11k12k13k14k15k16k21k22k23k24k25k26k31k32k33k34k35k36k41k42k43k44k45k46k51k52k53k54k55k56k61k62k63k64k65k66·a1a2a3a4a5a6,where VΔ denotes the volume of the finite element tetrahedron and the coefficients kij can be calculated from(12)kij=δiδj∇×wi→·∇×wj→.The element matrixMe of the second term from (7) can be written as(13)kA∑i=16∑j=16∫Δδiδjwi→·wj→aidV,where the dot product for edges m and n can be expanded as(14)wm→·wn→=Nim∇Njm-Njm∇Nim·Nin∇Njn-Njn∇Nin=NimNin∇Njm∇Njn-NimNjn∇Njm∇Nin-NjmNin∇Nim∇Njn+NjmNjn∇Nim∇Nin.The procedure for calculating the volume integrals from (14) can be found elsewhere [24].Finally, the element matrix[D]e of the right hand side from (7) can be written as(15)∑i=13∑j=13∫ΔdS→·wi→δi×wj→δjai,where the symbol Δ denotes the integration over triangle, and the vector product for edges m and n can be expanded as(16)wm→×wn→=NimNin∇Njm×∇Njn-NimNjn∇Njm×∇Nin-NjmNin∇Nim×∇Njn+NjmNjn∇Nim×∇Nin.The procedure for calculating the surface integrals from (16) can be found in [24].


### 2.6. SIE/MoM Approach for the Homogeneous Model

The human brain exposure can be estimated by means of coupled surface integral equations (SIE) [25, 26]:(17)jωμn∬SJ→r→′Gnr→,r→′dS′-jωεn∬S∇′S·J→r→′∇Gnr→,r→′dS′+∬SM→r→′×∇′Gnr→,r→′dS′=E→inc,n=10,n=2,where J→ and M→ are equivalent electric and magnetic current density, respectively, kn is the wave number of a medium nn=1,2, while Gn is the interior/exterior Green function for the homogeneous medium [25, 26]:(18)Gnr→,r→′=e-jknR4πR;R=r→-r→′.And R is the distance from the source to observation point.Set of integral equations (17) is solved by means of method of moments (MoM) procedure reported in [26]. The equivalent electric and magnetic currents J→ and M→ in (17) are expressed in terms of a linear combination of basis functions f→n and g→n, respectively.(19)J→r→=∑n=1NJnf→nr→,M→r→=∑n=1NMng→nr→,where Jn and Mn are unknown coefficients, while N is the total number of triangular elements.Applying the weighted residual approach, that is, multiplying (17) by the set of a test functions f→m and integrating over the surface S, after performing some mathematical manipulations, it follows that(20)jωμi∑n=1NJn∬Sf→mr→·∬S′f→nr→′GidS′dS+jωεi∑n=1NJn∬S∇S·f→mr→∬S′∇′S·f→nr→′GidS′dS±∑n=1NMn∬Sf→mr→·n^×g→nr→′dS+∑n=1NMn∬Sf→mr→·∬S′g→nr→′×∇′GidS′dS=∬Sf→mr→·E→incdS,i=10,i=2,where subscript i denotes the index of the medium. The details of the procedure can be found elsewhere (e.g., in [25, 26]).


## 2.1. Extracted (Single) Eye Model

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 Figure1.Figure 1
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 [11] while [12] reported spatial resolutions of 0.5×0.5×0.6 mm3. To the best of the authors knowledge, the most accurate MRI scans were performed using the 9.4 T magnet [13] achieving voxel volumes of about 0.13×0.13×0.8 mm3. However, in order to capture the fine details of the eye tissues such as choroid, retina, and iris, a much finer resolution is required. It is estimated that in order to have at least several layers of MRI voxels in the retina, one would need spatial MRI resolution of about 30 × 30 × 30 μm3 or less, or the resulting boundary surfaces between tissues would have become staircased.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 Figure2.Figure 2
Modeled tissues from the extracted and the compound eye models.The extracted eye model consists of 16 tissues, whose parameters are given in Table1. The frequency dependent dielectric parameters (the electrical conductivity σ and the relative permittivity ε) are modeled using the 4-Cole-Cole method [14]. Table 1 gives also the tissue mass density ρ.Table 1
Tissue parameters used in eye and head models.
Tissue
900 MHz
1800 MHz
ρ (g/m3)
σ (S/m)
ε (-)
σ (S/m)
ε (-)
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
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
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
1100The crystalline lens is modeled using five layers with varying relative permittivity, according to the Gradient Refraction Index (GRIN) model [15, 16].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.


## 2.2. Compound Eye and Head Model

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 Figure3(c). This model is named the compound eye model.Figure 3
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.
(a)
(b)
(c)The detailed model of the human head was constructed from the MRI of a 24-year-old male [17]. The current implementation of the model features 7 tissues in addition to 16 ocular tissues from the extracted eye model. The head tissue parameters are also modeled using the 4-Cole-Cole method [14] and are given in Table 1. The boundary surface of the complete head model is discretized using 5.934 triangular elements, while region inside the head is discretized using 4.073.250 tetrahedral elements.


## 2.3. Homogeneous Brain Model

As a very simple representation of the human brain, shown in Figure3(a), the homogenized realistic-shaped brain reported in [10] was used. To facilitate the preparation of otherwise a very complex organ, the freely available model from Google SketchUp was scaled to adult average brain using dimensions of length 167 mm, width 140 mm, height 93 mm, and volume of 1400 cm3 [18]. The surface of the model was discretized using the triangular elements, enabling the use of Rao-Wilton-Glisson (RWG) basis functions specially developed for triangular patches.Frequency dependent parameters of the homogeneous models are taken from [10]. In addition, linear and isotropic behavior is assumed for the electrical properties of tissues.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 [19].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.


## 2.4. Three-Compartment Head Model

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 [20], consisting of homogeneous compartments of brain, skull, and scalp, as depicted in Figure 3(b). This model is freely available and can be easily implemented in the BEM analysis [20]. The parameters of the three-shell model are given in Table 2.Table 2
Tissue parameters used in the three-compartment model.
Tissue
900 MHz
1800 MHz
ρ (g/m3)
σ (S/m)
ε (-)
σ (S/m)
ε (-)
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


## 2.5. Hybrid FEM/BEM Method

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 [9, 21]:(1)αE→ext′=E→inc′+∮∂Vn^×∇×E→extGdS+∮∂Vn^×E→ext×∇G+n^·E→ext∇GdS,where n^ is an outer normal to surface ∂V bounding the volume V and α is the solid angle subtended at the observation point. The total and the incident electric field are denoted by E→ext and E→inc, respectively, and G is the free space Green’s function given by(2)G=Gr→,r→′=e-jkR4πR;R=r→-r→′,where R is the distance from the observation point r→ to the source point r→′ and k denotes the wave number.Performing some manipulations, (1) can be specified in terms of tangential components of electric field E→ and magnetic field H→ on the boundary surface ∂V [9]:(3)αE→ext′=E→inc′-jωμ∮∂Vn^×H→extGdS+∮∂Vn^×E→ext×∇G-1σ+jωμ∇s·n^×H→ext∇GdS.The form of (3) is pertinent for coupling to the governing differential equations of the interior inhomogeneous domain, given by [9]:(4)∇×jωμ∇×E→int-σ+jωεE→int=0.The fieldsE→ and H→ are approximated using the edge elements [22] preserving the tangential continuity of the fields on the boundary(5)E→=∑i=1nδiw→iei,H→=∑i=1nδiw→ihi.The unknown coefficientsei and hi, respectively, associated with each edge of the model, are determined from the global system of equations while the coefficient δi=±1 is equal to 1 depending on whether the direction of local edge coincides with the chosen global edge direction.Vector base functionwk is given by [23] w→k=Ni∇Nj-Nj∇Ni, where Ni and Nj are the first order barycentric shape functions.After the weighted residual approach is applied to (4), followed by the dot product of (4) with test function wi, and using the Galerkin-Bubnov procedure, it can be written that(6)∫Vδiw→i·∇×jωμ∇×E→int-σ+jωεE→intdV=0.After applying some standard vector identities, followed by the divergence theorem, the weak form is obtained:(7)∫Vjωμ∇×δiw→i·E→int-σ+jωεδiw→i·E→intdV=∮∂VdS→·δiw→i×H→int.The FEM/BEM coupling can now be employed by using the continuity of the tangential components of electric and magnetic fields across the surface∂V. This leads to n^×E→int=n^×E→ext and n^×H→int=n^×H→ext. Thus, it can be written that(8)∮∂VdS→′·δiw→i×E→int′=∮∂VdS→′·δiw→i×E→ext′.After substitutingE→ext and H→ext in (3) with E→int and H→int, respectively, and after inserting (3) into (8), the following double surface integral is obtained:(9)∮∂VdS→′·δiw→i×αE→ext′=∮∂VdS→′·δiw→i×E→inc′-jωμ∮∂VdS→′·δiw→i×∮∂Vn^×H→intGdS+∮∂VdS→′·δiw→i×∮∂Vn^×E→int×∇GdS-1ω+jωμ∮∂VdS→′·δiw→i×∮∂V∇s·n^×H→ext∇GdS.Inserting (5) into (9) results in the system of equations related to the edges at the boundary surface of the problem:(10)Ebemebem=einc+HbemhbemEfemefem=Hfemhfem,where ebem and hbem are unknown coefficients associated with boundary surface of the scattering problem, einc are known coefficients calculated from the incident field, and matrices [Ebem] and [Hbem] arise from boundary integral equation (9), while matrices [Efem] and [Hfem] stem from FEM (7).After having discretized the interior domain into finite set of tetrahedrons, and using the expansion given by (5), followed by the Galerkin procedure, the finite element matrices could be written. Thus, the element matrix Ke of the first term from (7), after some work, can be written as(11)1kB∑i=16∑j=16∇×wi→δi·∇×wj→δjaidV=VΔkBk11k12k13k14k15k16k21k22k23k24k25k26k31k32k33k34k35k36k41k42k43k44k45k46k51k52k53k54k55k56k61k62k63k64k65k66·a1a2a3a4a5a6,where VΔ denotes the volume of the finite element tetrahedron and the coefficients kij can be calculated from(12)kij=δiδj∇×wi→·∇×wj→.The element matrixMe of the second term from (7) can be written as(13)kA∑i=16∑j=16∫Δδiδjwi→·wj→aidV,where the dot product for edges m and n can be expanded as(14)wm→·wn→=Nim∇Njm-Njm∇Nim·Nin∇Njn-Njn∇Nin=NimNin∇Njm∇Njn-NimNjn∇Njm∇Nin-NjmNin∇Nim∇Njn+NjmNjn∇Nim∇Nin.The procedure for calculating the volume integrals from (14) can be found elsewhere [24].Finally, the element matrix[D]e of the right hand side from (7) can be written as(15)∑i=13∑j=13∫ΔdS→·wi→δi×wj→δjai,where the symbol Δ denotes the integration over triangle, and the vector product for edges m and n can be expanded as(16)wm→×wn→=NimNin∇Njm×∇Njn-NimNjn∇Njm×∇Nin-NjmNin∇Nim×∇Njn+NjmNjn∇Nim×∇Nin.The procedure for calculating the surface integrals from (16) can be found in [24].


## 2.6. SIE/MoM Approach for the Homogeneous Model

The human brain exposure can be estimated by means of coupled surface integral equations (SIE) [25, 26]:(17)jωμn∬SJ→r→′Gnr→,r→′dS′-jωεn∬S∇′S·J→r→′∇Gnr→,r→′dS′+∬SM→r→′×∇′Gnr→,r→′dS′=E→inc,n=10,n=2,where J→ and M→ are equivalent electric and magnetic current density, respectively, kn is the wave number of a medium nn=1,2, while Gn is the interior/exterior Green function for the homogeneous medium [25, 26]:(18)Gnr→,r→′=e-jknR4πR;R=r→-r→′.And R is the distance from the source to observation point.Set of integral equations (17) is solved by means of method of moments (MoM) procedure reported in [26]. The equivalent electric and magnetic currents J→ and M→ in (17) are expressed in terms of a linear combination of basis functions f→n and g→n, respectively.(19)J→r→=∑n=1NJnf→nr→,M→r→=∑n=1NMng→nr→,where Jn and Mn are unknown coefficients, while N is the total number of triangular elements.Applying the weighted residual approach, that is, multiplying (17) by the set of a test functions f→m and integrating over the surface S, after performing some mathematical manipulations, it follows that(20)jωμi∑n=1NJn∬Sf→mr→·∬S′f→nr→′GidS′dS+jωεi∑n=1NJn∬S∇S·f→mr→∬S′∇′S·f→nr→′GidS′dS±∑n=1NMn∬Sf→mr→·n^×g→nr→′dS+∑n=1NMn∬Sf→mr→·∬S′g→nr→′×∇′GidS′dS=∬Sf→mr→·E→incdS,i=10,i=2,where subscript i denotes the index of the medium. The details of the procedure can be found elsewhere (e.g., in [25, 26]).


## 3. Results and Discussion

### 3.1. Human Eye Models

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 Figures4–7. The incident plane wave of 1 GHz and 1800 MHz is horizontally and vertically polarized and directed toward the corneal surface, perpendicular to the coronal head/eye cross-section. The amplitude of the incident EM wave is taken to be 1 V/m.Figure 4
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.
(a)
(b)
(c)
(d)Figure 5
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.
(a)
(b)
(c)
(d)Figure 6
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.
(a)
(b)
(c)
(d)Figure 7
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.
(a)
(b)Figure4 shows the results for the induced field on the surface of the extracted eye model and the compound eye model, respectively, due to incident wave of 1 GHz. From Figure 4 it is obvious that the distribution of electric field on the surface of the compound eye model (cornea and sclera) is not showing the symmetrical nature as is the case for the extracted eye model. In addition, the obtained peak values in the compound model are higher than the respected values in the extracted model, in addition to being different for both polarizations, again, contrary to the extracted model where the same values were obtained. These results suggest the significant influence of the neighboring head tissues on the induced electric field values.The similar finding was shown for the 1800 MHz case, as shown in Figure5, although the discrepancy in the maximum values between the two models is significantly less.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 Figures6 and 7 where the results for the induced field on the transverse cross-sections of the eye models and the electric field distribution along the eye visual axis, respectively, are shown.The results from the previous study [8], obtained for vertically polarized wave at 1 GHz, showed similar distribution of the induced electric field along the pupillary axis in both compound and extracted eye models, as seen again in Figure 7(a), suggesting the usefulness of the extracted eye model in the initial EM exposure assessment. However, the results for horizontal polarization showed higher discrepancy. Moreover, as seen in Figure 7(b), this is even more pronounced at 1800 MHz, for both polarizations.One interesting fact that can be seen from Figure7 is that, due to the symmetrical nature of the employed eye model, the extracted model basically gives the same results for both polarizations, as shown by almost ideally overlapping graphs, suggesting that this model does not discriminate between the two polarizations of the incident plane wave.


### 3.2. Human Brain Models

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 Figures8–11. Although a similar distribution of the electric field was obtained, as shown by the colormaps, the lower maximum values were obtained in all four cases in the homogeneous brain model compared to the other two brain models.Figure 8
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.
(a)
(b)
(c)Figure 9
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.
(a)
(b)
(c)Figure 10
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.
(a)
(b)
(c)Figure 11
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.
(a)
(b)
(c)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 theT=696 triangular elements and N=1044 edge elements. The reason for such a low number is that employed SIE formulation results in fully populated matrices, thus preventing the current implementation from running on large system matrices. The low number of elements will also result in some numerical artifacts particularly evident in the superficial areas of the model, evident as peaks in Figures 12–15.Figure 12
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.Figure 13
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.Figure 14
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.Figure 15
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 Figures8–11. Although the surface of three-compartment model is rather smoothed, lacking a cortex details featured in the conformed model, similar maximum values were obtained.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 Figures12–15. In general, similar trends are obtained in all models; however, only the detailed compound model shows the wavelike behavior in the interior domain.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.


## 3.1. Human Eye Models

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 Figures4–7. The incident plane wave of 1 GHz and 1800 MHz is horizontally and vertically polarized and directed toward the corneal surface, perpendicular to the coronal head/eye cross-section. The amplitude of the incident EM wave is taken to be 1 V/m.Figure 4
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.
(a)
(b)
(c)
(d)Figure 5
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.
(a)
(b)
(c)
(d)Figure 6
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.
(a)
(b)
(c)
(d)Figure 7
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.
(a)
(b)Figure4 shows the results for the induced field on the surface of the extracted eye model and the compound eye model, respectively, due to incident wave of 1 GHz. From Figure 4 it is obvious that the distribution of electric field on the surface of the compound eye model (cornea and sclera) is not showing the symmetrical nature as is the case for the extracted eye model. In addition, the obtained peak values in the compound model are higher than the respected values in the extracted model, in addition to being different for both polarizations, again, contrary to the extracted model where the same values were obtained. These results suggest the significant influence of the neighboring head tissues on the induced electric field values.The similar finding was shown for the 1800 MHz case, as shown in Figure5, although the discrepancy in the maximum values between the two models is significantly less.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 Figures6 and 7 where the results for the induced field on the transverse cross-sections of the eye models and the electric field distribution along the eye visual axis, respectively, are shown.The results from the previous study [8], obtained for vertically polarized wave at 1 GHz, showed similar distribution of the induced electric field along the pupillary axis in both compound and extracted eye models, as seen again in Figure 7(a), suggesting the usefulness of the extracted eye model in the initial EM exposure assessment. However, the results for horizontal polarization showed higher discrepancy. Moreover, as seen in Figure 7(b), this is even more pronounced at 1800 MHz, for both polarizations.One interesting fact that can be seen from Figure7 is that, due to the symmetrical nature of the employed eye model, the extracted model basically gives the same results for both polarizations, as shown by almost ideally overlapping graphs, suggesting that this model does not discriminate between the two polarizations of the incident plane wave.


## 3.2. Human Brain Models

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 Figures8–11. Although a similar distribution of the electric field was obtained, as shown by the colormaps, the lower maximum values were obtained in all four cases in the homogeneous brain model compared to the other two brain models.Figure 8
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.
(a)
(b)
(c)Figure 9
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.
(a)
(b)
(c)Figure 10
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.
(a)
(b)
(c)Figure 11
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.
(a)
(b)
(c)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 theT=696 triangular elements and N=1044 edge elements. The reason for such a low number is that employed SIE formulation results in fully populated matrices, thus preventing the current implementation from running on large system matrices. The low number of elements will also result in some numerical artifacts particularly evident in the superficial areas of the model, evident as peaks in Figures 12–15.Figure 12
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.Figure 13
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.Figure 14
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.Figure 15
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 Figures8–11. Although the surface of three-compartment model is rather smoothed, lacking a cortex details featured in the conformed model, similar maximum values were obtained.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 Figures12–15. In general, similar trends are obtained in all models; however, only the detailed compound model shows the wavelike behavior in the interior domain.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.


## 4. Conclusion

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.



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*Source: 7932604-2017-11-26.xml*