A theoretical model for the frequency-dependent dielectric properties of corneal tissue at microwave frequencies.
Dielectric properties of biological tissues have been of interest for several reasons. They provide physical measures that are closely related to the underlying microstructure and chemical composition, can be used as indicators for non-invasive diagnosis and medical sensing [1-3], and are useful for computational dosimetry of electromagnetic waves [4-8] or computational evaluation of propagation in tissues for communication with biomedical devices [9,10]. Data on dielectric properties can be obtained through spectroscopic measurements . As recently shown [12,13], dielectric properties of tissues can also be theoretically estimated and reproduced by considering the microstructure and material composition of the tissue, and quasi-electrostatic field solutions. Theoretical models are independent of direct spectroscopic measurements, and rely on appropriately combining the known properties of different, basic constituents (which is known as "Mixing"). Walker et al.  used a theoretical tissue model for cervical squamous tissue. A theoretical skin model has been developed in the MHz range by Huclova et al.  for non-invasive sensing, based on the effective medium approach, mixing formulas and numerical analysis. Theoretical models are extremely useful in noninvasive medical sensing, because they provide a connection between tissue structure and dielectric response. Pathological changes in tissue microstructure (e.g., cancer) cause changes in the dielectric response, which can be interpreted based on the theoretical model .
Recently, studies of radiation safety are in progress which involve computational evaluation of THz absorption in superficial tissues; i.e., the skin and the eyes (esp. the cornea) . As is well-known, experimental data on dielectric properties of tissues are rather scarce in this frequency region. In such applications, a theoretical model might be the most promising option for estimating the dielectric properties of tissues and their substructures. Obviously, once such a model is developed, it should be verified using the existing experimental data at lower frequencies (e.g., microwaves).
Theoretical models of skin have been reported in the RF frequency range . To the best of our knowledge, a theoretical electromagnetic model for the cornea has not yet been developed. Applications for this model can be anticipated in computational dosimetry and non-invasive sensing [14, 15].
We provide, for the first time, a theoretical electromagnetic model for predicting the electrical properties of the corneal tissue based on its physical microstructure and chemical composition. Interestingly as we shall see, the cornea is quite amenable to theoretical analysis for two reasons: Because of its role in vision, the cornea is a highly ordered tissue. This fact facilitates modeling by relatively simple mixture rules. Moreover, corneal hydration (water content) is actively and strictly controlled in the eye and is kept almost constant in a healthy tissue. We thus expect good comparison with experimental datasets to be feasible within a reasonable uncertainty. These features are not present in the case of skin, where notable differences arise between wet and dry states.
Objective: Our purpose is to develop a theoretical cornea model with sufficient complexity to agree within [+ or -]20% or better with available measured values in the microwave region.
Perhaps the most extensively used set of experimental tissue data to date are those of Gabriel et al., compiled in  based on the works of Gabriel et al. [16,17] up to 100GHz. In the early 2000s, a group of researchers including C. Gabriel performed new dielectric measurements in vivo and in vitro as well as a literature review . As concerns the cornea, the experimental results were close to the previous results of Gabriel et al. [17, 19].
In this paper, we use Cole-Cole and Debye expressions to express the complex frequency-dependent dielectric properties of basic constituents such as water. Many dielectric relaxation processes in basic materials can be meaningfully described by the Cole-Cole expression for the complex permittivity [epsilon], as in Eq. (1):
[epsilon] = [[epsilon].sub.[infinity]] + [[DELTA][epsilon]/1 + [(j2[pi]f[tau]).sup.1-[alpha]]] + [[[sigma].sub.i]/j[omega][[epsilon].sub.0]] (1)
where [DELTA][epsilon] = ([[epsilon].sub.s] - [[epsilon].sub.[infinity]]). The expression in (1) reproduces one relaxation from a higher permittivity [[epsilon].sub.s] at lower frequencies to a lower permittivity [[epsilon].sub.[infinity]] at higher frequencies, with the transition taking place at the relaxation frequency f = [(2[pi][tau]).sup.-1]. The parameter [alpha] varies between 0 and 1, with higher values expressing broader dispersions. The special case of [alpha] = 0 corresponds to a Debye expression with an additional conductivity term. The symbol [[sigma].sub.i] stands for ionic conductivity . The ionic conductivity is more important at lower frequencies.
In the following, Section 2 introduces the relevant structural information about the cornea. Section 3 presents the procedure of theoretical modeling. Section 4 then follows with comparison with experimental data and discussion.
2. THE STRUCTURE AND COMPOSITION OF CORNEA
The cornea is a living tissue in the outer surface of the eyeball. In humans it has a thickness of almost 0.5 millimeter, and comprises five discernible layers. These are, from the outermost to the innermost, the epithelium; Bowman's membrane; the stroma; Descemet's membrane, and the endothelium . These are graphically reproduced in Figure 1(a). The epithelium and endothelium layers consist of tightly packed cells (Table 1). In humans the epithelium consists of approximately 6 layers of cells, whereas the 5 [micro]m thick endothelium layer consists of a single layer of cells only . The Bowman's and Descemet's membranes are relatively thin, tough layers, rich in collagen fibrils  and both about 10-20 [micro]m thick. Collagen is the name of a family of organic molecules that can provide for high tensile strength by forming very long fibrils . The stroma can be described as being composed of collagen fibrils in an aqueous background (the ground substance). A sparse distribution (a volume fraction of about 12%) of cells (keratocytes, Figure 1(b)), rest between the lamellae . The ground substance of the stroma is itself composed of water and other non-fibrilar components . The stroma is of dominant importance in our treatment here, since it constitutes 90% of the whole cornea .
A closer study of the stroma reveals that long collagen fibrils run across the stroma as shown in Figure 1(b) . One can consider the stroma as composed of stacked lamellae, where the orientations of the fibrils in each lamella are identical, but at specific angles with respect to adjacent lamellae. Specifically, fibrilar orientations from adjacent lamellae tend to be at right angles to each other in the central region of the cornea , as shown in Figure 1(b). The lamellae might branch and interweave at some points. Interweaving of the lamellae has been ignored in the model.
In terms of composition, Table 1 shows the dimensions of the fibrils, volume fractions, and other geometrical parameters we shall use in this study. In Table 1, we observe the similarity of collagen volume fractions for human and porcine tissues. As Meek and Leonard  show in their experimental study of cornea ultrastructure in 45 types of species, the fibrilar volume fraction, and intra-fibrilar spacing between collagen molecules is almost constant among mammals; a fact which is further justified by considering the small variation of the optical refractive index of the cornea among species; namely [n.sub.cor] = 1-372 [+ or -] 0.009 .
3. THE CORNEA MODEL
3.1. Preliminary Considerations
The idealized model we shall use for the cornea is depicted in Figure 2. The Bowman's and Descemet's membranes have been merged into the model for the stroma. This is justified by the similar collagen-based composition of these layers when compared to the stroma. Furthermore, it has been shown experimentally in  that samples of stroma with or without the Descemet's membrane have almost the same permeability to ions and several larger molecules, indicating that the Descemet's membrane is actually not an electrically "impeding" membrane, as is observed, e.g., with cell membranes.
We note that the structure is anisotropic. With reference to the coordinates in Figure 2, it is desired to calculate the effective dielectric tensor [[bar.[epsilon]].sub.cor]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where in order to facilitate the presentation, we introduce the following notational convention for field polarizations and their corresponding permittivities: superscripts [[parallel].sub.l] and [[perpendicular to].sub.l] indicate directions parallel or perpendicular to the planar lamellae (or to the surface of the cornea), respectively, and superscripts [[parallel].sub.f] and [[perpendicular to].sub.f] indicate direction parallel or perpendicular to linear collagen fibrils, respectively. Subscripts describe the medium to which the effective permittivity is assigned, and are defined where they first appear throughout the paper.
Practically, the process of computing the overall dielectric properties of the tissue (mixing) corresponds to replacing the heterogeneous material with a "homogenized" model, which has the same overall dielectric behaviour. There are numerous "mixture" rules that are derived for different geometries from rigorous field solutions. Mixture rules differ from simple spatial averaging of permittivity, unless for some special cases. A review of many useful mixture rules, including those used in this paper, can be found in . We use the mixture rules with complex, frequency-dependent dielectric parameters as in . In this way, we also incorporate material dispersions into the model.
Considering the cornea, one can distinguish several "scales" of electrical heterogeneity. The tissue is heterogeneous in the molecular, cellular, and also in the macroscopic tissue scales. The theoretical modeling of dielectric properties for the tissue begins by starting from the smallest scale, applying mixture rules, replacing the heterogeneity in that scale with a homogeneous model, and proceeding to the next, larger scale, where one has the homogenized model of the previous step among other constituents of the larger scale. The process of using mixture rules can now be repeated as in the previous step. The mixing steps are schematically shown in Figure 3 for the cornea.
At each scale, the process of mixing and the effective properties obtained are valid as long as the inclusions remain small with respect to the wavelength. We consider electrically small to mean smaller than [[lambda].sub.w]/10 or [[lambda].sub.w]/5, with [[lambda].sub.w] denoting the wavelength inside an aqueous background (Table 2). As a result, the cornea model with a stroma of 500 | m thickness is physically meaningful up to about 20 GHz. The consecutive steps in Figure 3 are now briefly described.
3.2. The Model
3.2.1. The Aqueous Ground Substance
To estimate the complex permittivity of the ground substance in the stroma, first the ionic content in the ground substance is modeled by the addition of a conductivity term to the water model. The water model [[epsilon].sub.w](f) is given in Table 2. This yields the aqueous electrolyte permittivity [[epsilon].sub.lyt]:
[[epsilon].sub.lyt] = [[epsilon].sub.w] + [[[sigma].sub.i]/j[omega][[epsilon].sub.0]] (3)
The value of [[sigma].sub.i] for most biological electrolytes is estimated to be on the order of 1S x [m.sup.-1] (an example is discussed by Gimsa et al. ). We have used [[sigma].sub.i] = 1.35 S x [m.sup.-1]. This value is close to [[sigma].sub.i] = 1.2 S x [m.sup.-1] mentioned by Huclova et al.  for biological electrolytes and provides slightly better comparison with experimental data.
Proteins and other macromolecules account for much of the non-fibrilar material volume fraction. The final permittivity of the ground substance is obtained by applying the Maxwell-Garnet (MG) Mixture rule, which assumes spherical particles of [[epsilon].sub.nf] in a background of [[epsilon].sub.lyt]  and by using the volume fraction of non-fibrilar material [[upsilon].sub.nf] as given in Table 1.
[[epsilon].sub.gs] = [[epsilon].sub.lyt] + 3[[upsilon].sub.nf][[epsilon].sub.lyt] [[[epsilon].sub.nf] - [[epsilon].sub.lyt]/[[epsilon].sub.nf] + 2[[epsilon].sub.lyt] - [[upsilon].sub.nf]([[epsilon].sub.nf] - [[epsilon].sub.lyt])]. (4)
We note that although the inclusions are of molecular dimensions, their effect can nevertheless be reasonably theoretically estimated by appropriate mixture rules according to the literature [30, 31]. Although it has been shown that the Bruggeman mixture formula can provide more accurate results than MG , our experience showed only minor difference between these formulas for the relatively low volume fractions here.
3.2.2. The Lamellae
Every lamella is composed of the aqueous ground substance and collagen fibrils as inclusions. We shall distinguish between two cases, i.e., where the electrical field vector is perpendicular to the fibrils, giving [[epsilon].sup.[perpendicular to]f.sub.l] and where it parallels the fibrils, resulting in [e.sup.[parallel]f.sub.l]. We thus use the MG mixture formula for cylindrical inclusions  for [[epsilon].sup.[perpendicular to]f.sub.l]. The linear mixture law , which is the well-known formula for combination of parallel capacitors, is used to find [[epsilon].sup.[parallel]f.sub.l]
[[epsilon].sup.[perpendicular to]f.sub.l] = [[epsilon].sub.gs] + 2[[upsilon].sub.c][[epsilon].sub.gs] [[[epsilon].sub.c] - [[epsilon].sub.gs]/[[epsilon].sub.c] + [[epsilon].sub.gs] - [[epsilon].sub.c]([[epsilon].sub.c] - [[epsilon].sub.gs])]. (5)
[[epsilon].sup.[parallel]f.sub.l] = [[epsilon].sub.gs] x (1 - [[upsilon].sub.c]) + [[epsilon].sub.c] x [[upsilon].sub.c] (6)
3.2.3. The Stroma
With reference to Figure 2, the field component perpendicular to the lamellae [E.sup.[perpendicular to]l] = [E.sub.z] is always perpendicular to the fibrils, thus for [E.sup.[perpendicular to]l] all lamellae appear with the same permittivity [[epsilon].sup.[perpendicular to]f.sub.l]. We thus conclude that [[epsilon].sup.[perpendicular to]l.sub.s] = [[epsilon].sup.[perpendicular to]f.sub.l]. For [E.sup.[parallel]l] (e.g., assuming either [E.sub.x] or [E.sub.y] in Figure 2) fibril orientations appear periodically parallel or perpendicular to the field, and we can use the linear mixture rule as in Eq. (6) to find the effective permittivity.
[[epsilon].sup.[parallel]l.sub.s] = 1/2 ([[epsilon].sup.[parallel]f.sub.l] + [[epsilon].sup.[perpendicular to]f.sub.l]). (7)
[[epsilon].sup.[perpendicular to]l.sub.s] = [[epsilon].sup.[perpendicular to]f.sub.l] (8)
To model the keratocytes dispersed throughout the stroma, the MG mixture rule for double layer spherical inclusions [28, Eq. (3.32)] is used with the volume fraction [[upsilon].sub.k] and membrane ([[epsilon].sub.m]) and cytoplasm ([[epsilon].sub.cyto]) properties as described in Table 3. The cytoplasm is modeled based on the frequency-dependent cytoplasm parameters from Gimsa etal. . specifically using the expression from Table 2 with [[epsilon].sub.1] = 212, [[epsilon].sub.2] = 50, [[epsilon].sub.[infinity]] = 4, [[tau].sub.1] = 10.61 ns, [[tau].sub.2] = 7.2 ps and [[sigma].sub.i] = 1 S x [m.sup.-1] with a Cole-Cole broadening factor of [[alpha].sub.1] = 0.5 (Eq. (1)). Letting [[epsilon].sub.sk] denote the complex permittivity of the stroma with keratocytes,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where w = [[(R - [d.sub.m])/R].sup.3] with R denoting the cell radius and [d.sub.m] the membrane thickness from Table 3. The value for R has been chosen equal to half the keratocyte thickness (Table 3). In Eq. (9), [[epsilon].sup.[perpendicular to]l.sub.sk] and [[epsilon].sup.[parallel]l.sub.sk] will be obtained, if [[epsilon].sub.s] is substituted by [[epsilon].sup.[perpendicular to]l.sub.s] and [[epsilon].sup.[parallel]l.sub.s], respectively.
3.2.4. The Cornea Model
Finally, for the last step in the model hierarchy, the stromal layer has to be considered as enclosed between the cellular epithelium and endothelium layers. These are modeled by cytoplasm ([[epsilon].sub.cyto]) and membranes ([[epsilon].sub.m]) of thickness [d.sub.m]. For [[epsilon].sup.[perpendicular to]l.sub.cor], the membranes parallel to the field can be neglected because of their extremely low volume fraction. The cell membranes perpendicular to the field orientation are considered because of their probable blocking effect against ionic currents (the well-known capacitive effect at poorly conducting membranes) at lower frequencies. These are then mixed by the well-known series capacitors formula; Eq. (10). The parameter [N.sub.m] = 14 is the number of membrane layers, 12 due to 6 rows of epithelial cells and 2 due to one row of endothelial cells.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
For [[epsilon].sup.[parallel]l.sub.cor], we first take into account the horizontal membranes in Figure 2, by defining [[epsilon].sub.en,ep] calculated by a series-capacitance rule Eq. (11), followed by a linear mixing rule in Eq. (12).
[[epsilon].sub.en,ep] = ([d.sub.cyto] + 2[d.sub.m]) x [([[d.sub.cyto]/[[epsilon].sub.cyt]] + [2 x [d.sub.m]/[[epsilon].sub.m]]).sup.-1] (11)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
The set of Formulas (3)-(12) constitutes a model for the cornea composed of smaller individual models.
4. RESULTS AND DISCUSSION
We compare the whole tissue permittivity and conductivity obtained from our model with experimental data models from the literature. The model parameters for porcine cornea from Table 1 are used to provide comparison with a recent set of cornea data  measured and reported for three age-groups of pigs over the 0.4-10 GHz frequency range. In accordance with the measurement procedures  which correspond to the parallel electric field polarization for the cornea, comparison is presented for [[epsilon].sup.[parallel]l.sub.cor] in Figure 4 and in the rest of the paper. The model shows agreement with the experimental data within the usual experimental tolerance of 20%.
We now investigate a possible further refinement for the model. In Section 3, we used a free-water model in our analysis. A slight decrease in relaxation frequency in the experimental data of Figure 4 from that predicted by the theoretical model (dashed line) suggests the presence of bound-water, which is usually slower than free water. We model the presence of bound water as in  by considering a certain volume fraction of water to have a principal relaxation time [[tau].sub.h] instead of [[tau].sub.1] in Table 2. The model with bound water can have its relative difference reduced to within 5% (3% for one of the datasets) for the real part and to within nearly 17% (10% for one of the datasets) for the conductivity, well within the usual experimental uncertainty (Figures 4, 5). This model assumes a volume fraction of bound water equal to [[upsilon].sub.b] = 0.3 and a relaxation time of [[tau].sub.h] = 4[[tau].sub.1], which are reasonable values according to the literature .
Consequently as understood from Figure 5, inclusion of bound-water in the model is promising and seems to provide even better agreement especially at higher frequencies. However, it has not been included in the theoretical model of the previous section, mostly due to the uncertainties involved in related parameters. More detailed knowledge of bound water properties in the cornea can be used to refine the model in future.
Further comparison with experimental data is provided in the online supplement for both polarizations, and shows agreement within 20%. It is thus observed that the model reproduces the experimentally observed behaviour above several hundred MHz very well. The model has good accuracy also in the microwave region, which has been the target of this work.
In order to assess the meaningfulness of the model and the significance of its agreement with measured values, and to understand the model dependence on parameter values, we have also performed a sensitivity analysis for [[epsilon].sup.[parallel]l.sub.cor](f). By defining the sensitivity S as the "ratio of relative changes", [[epsilon].sub.cor] (1 GHz) shows significant sensitivity to [[upsilon].sub.c], (S = -0.14), and [[upsilon].sub.nf], (S = -0.12), but shows insignificant dependence on [[epsilon].sub.c] and [[epsilon].sub.nf] (S = 0.01) and [[sigma].sub.i] (S = 0.00). This can be understood by the relatively high dielectric permittivity of water at lower frequencies. Low-permittivity inclusions such as collagen occupy a certain volume, excluding a corresponding amount of water. This leads to a decrease in permittivity which is more sensitive to the excluded water volume, and less sensitive to the actual permittivity of the inclusions. Obviously, [[sigma].sub.cor] (1 GHz) depends strongly on ionic content (S = 0.65), but this dependence fades at higher frequencies, where the dielectric loss of water becomes dominant. Interestingly, as several high sensitivity values indicate, the good agreement is not trivial, but rather meaningfully related to the values chosen for the model parameters. A set of calculated sensitivity values for several model parameters is provided in the online supplement. The supplement also includes a description of further attempts at numerical verification of the cornea model.
One of the advantages of the theoretical model is its independence from measurements; it is not a mathematical expression fitted to experimental data, but rather a physics-based bottom-to-top method for theoretically reproducing a set of tissue properties over a wide range of frequencies. Currently, refinement of the model using the aforementioned ideas, using the model for predicting the dielectric properties of corneal tissue sub-structures at higher frequencies for computational dosimetry, and some ophthalmological applications are under investigation.
We have presented a fine model for theoretical prediction of the complex permittivity of the cornea based on its material composition and by applying appropriate mixing rules to the microstructure of the cornea in several steps. Good agreement of the theoretical model predictions with reported experimental models in the literature have been obtained at microwave frequencies within the target accuracy of 20%. Sensitivity analysis for the model parameters have been performed, which provides support for the physical meaningfulness of the model. It has been shown that the model can be refined at higher frequencies by considering the effect of bound water. The model is expected to find application in non-invasive medical sensing and prediction of dielectric properties for high-frequency computational dosimetry.
M. Saviz expresses gratitude to Prof. Dr. M. Clemens, head of the Chair of Electromagnetic Theory at the University of Wuppertal for facilitating this research. Dr.-Ing. J. Streckert has kindly and carefully reviewed the manuscript and pointed out many improvements. Constructive discussions with Dr.-Ing. D. N. Chigrin are warmly appreciated. Prof. Dr.-Ing. V. Hansen has been supporting this work with helpful experience and encouragement. M. Sc. O. Spathmann has been helpful with the literature regarding experimental data. The work of M. Saviz is supported by a Grant from Iran Telecommunications Research Center.
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Mehrdad Saviz (1), * and Reza Faraji-Dana (2)
(1) Antenna Laboratory, School of Electrical and Computer Engineering, University of Tehran, P.O. Box 14395-515, North Kargar Avenue, Tehran, Iran
(2) Center of Excellence on Applied Electromagnetic Systems, School of Electrical & Computer Engineering, University of Tehran, P.O. Box 14395-515, North Kargar Avenue, Tehran, Iran
Received 25 November 2012, Accepted 11 January 2013, Scheduled 26 February 2013
* Corresponding author: Mehrdad Saviz (email@example.com).
Table 1. Model parameters. Symbol Description Value (Human) [d.sub.epi] Thickness of 50 [micro]um the epithelium [d.sub.endo] Thickness of 5 [micro]um the endothelium [d.sub.s] Thickness of 500 [micro]um the stroma [d.sub.cornea] Corneal ~ 555 [micro]um thickness [[upsilon.sub.c] Collagen volume 0.124 fraction in stroma [[upsilon.sub.nf] Volume fraction of non-fibrilar 0.104 material from ground substance -- Diameter of a 30.8 nm collagen fibril -- Interfibrilar 55.3 [+ or -] distance 4.00 nm [[epsilon].sub.c] Permittivity estimation 2.4 for collagen [[epsilon.sub.nf] Permittivity approximation for non-fibrilar material in the 3 (2-5) stromal ground substance (ions, macromolecules) [[upsilon]. Mean volume sub.kerato] fraction of 0.122 keratocytes in the stroma Symbol Value References (Porcine) [d.sub.epi] 80 [micro]m [20,38] [d.sub.endo] 8[micro]m [38, 39] [d.sub.s] 800 [micro]m  and [20, 38] [d.sub.cornea] ~ 888 um [20, 38] [[upsilon.sub.c] 0.124  [[upsilon.sub.nf] 0.104  -- 36.9 [+ or -]  3.2 nm -- 58.6 [+ or -]  4.5 nm [[epsilon].sub.c] 2.4  [[epsilon.sub.nf] 3 (2-5) [30, 31] [[upsilon]. sub.kerato] 0.111 [23, 42] Table 2. The double-Debye dielectric model of water . Equation [[epsilon].sub.w] = [[epsilon].sub.[infinity]] + [([[epsilon].sub.1] - [[epsilon].sub.2])/[1 + j2[pi]f[[tau].sub.1]]] + [([[epsilon].sub.2] - [[epsilon].sub.[infinity]])/[1 + 2[pi]f[[tau].sub.2]]] Symbol Description Value [[epsilon].sub.1] Static permittivity (37[degrees]C) 74.23 [[epsilon].sub. Limit permittivity at high frequencies 3.76 [infinity]] [[epsilon].sub.2] Intermediate permittivity 4.98 [[tau].sub.1] Time constant of the first relaxation 6.26 ps [[tau].sub.2] Time constant of the second relaxation 0.157ps Table 3. Parameters for the cellular components (epithelium, endothelium, keratocytes). Symbol Description Value References [d.sub.m] Thickness of the 5.5 nm  membrane [d.sub.cyto] Average cell 5 [micro]m  diameter in the epi/endothelium [d.sub.kerato] Keratocyte body 30.9 [micro]m  length [t.sub.kerato] Keratocyte 1.34 [micro]m  thickness [[epsilon].sub.m] Permittivity of 9.0  lipid membrane models [[epsilon].sub.cyt] Complex See text  permittivity of the cytoplasm
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|Author:||Saviz, Mehrdad; Faraji-Dana, Reza|
|Publication:||Progress In Electromagnetics Research|
|Date:||May 1, 2013|
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