Analytical and Numerical Models for TE-Wave Absorption in a Graded-Index GNP-Treated Cell Substrate Inserted in a Waveguide

Abstract

In this paper, absorption phenomena in a hollow waveguide with an inserted graded dielectric layer are studied, for the case of transverse electric (TE) wave propagation. The waveguide model aims to be applicable to a study of a potential cancer treatment by heating of gold nanoparticles (GNPs) inside the cancer cells. In our previous work, general exact analytical fomulas for transmission, reflection, and absorption coefficients were derived. These fomulas are further developed here to be readily applicable to the calculation of the absorption coefficient within the inserted lossy layer only, quantifying the absorption in the GNP-fed cancer tissue. To this end, we define new exact analytic scale factors that eliminate unessential absorption in the surrounding lossy medium. In addition, a numerical model was developed using finite element method software. We compare the numerical results for power transmission, reflection and absorption coefficients to the corresponding results obtained from the new modified exact analytic fomulas. The study includes both a simple example of constant complex permittivities, and a more realistic example where a dispersive model of permittivity is used to describe human tissue and the electrophoretic motion of charged GNPs. The results of the numerical study with both non-dispersive and dispersive permittivities indicate an excellent agreement with the corresponding analytical results. Thus, the model provides a valuable analytical and numerical tool for future research on absorption phenomena in GNP-fed cancer tissue.

1. Introduction

The heating of biological tissue with inserted gold nanoparticles (GNPs) by radiofrequency (RF) electromagnetic radiation is an important phenomenon with potential to be used as a non-invasive method to treat cancer [1,2]. The method relies on the cancer cells’ property to intake an abnormal amount of nutrients. If GNPs are coated with suitable nutrient ligands, they can be taken into the cancer cells with high concentration, while their concentration in normal cells remains low [3,4,5]. Consequently, the temperature of cells with high GNP concentration will increase to such levels to cause the death of such (cancer) cells under exposure of an externally applied RF field. This effect occurs at temperatures between 43 and 49 °C [6,7,8,9]. Furthermore, normal cells recover faster than cancer cells when exposed to heat, due to having more blood flow and hence greater heat dissipation [10]. In addition, hyperthermia in this context seems to actually enhance the immune system instead of harming it [9]. The ligand shells also provide the GNPs with a net charge [11,12]. The size of the GNPs cannot be larger than 5 nm in order for them to pass through the kidneys [13].

While the physical mechanism behind the heating of GNPs under optical frequencies is well understood to be caused by localized surface plasmon resonances [14,15,16], the mechanism behind RF heating of GNP-fed biological tissue has, however, been a subject to extensive investigation. It is established that the main source of cell heating is the electrophoretic acceleration of charged particles, while Joule and inductive heating are found to be negligible [2,11,17]. The applied RF radiation drives the GNPs into resonant oscillation leading to dielectric losses associated with the electrophoretic current. A challenge for the proposed medical application is that absorption occurs in both the target cancer cells and in the background media. This was investigated in previous work [18,19] for a spherical/ellipsoidal suspension of GNPs in a spatially unlimited system. An alternative approach is to isolate the system in a waveguide setting as shown in more recent work [20,21] for the purpose of increased control of the environment and as a practical comparison to future experimental measurements.

The overall objective of the research reported in the present paper is to develop theoretical and practical techniques to identify GNP model parameters, and obtain optimal electrophoretic resonance in a thin GNP-treated cell substrate inserted in a waveguide. This is shown in the overview in Figure 1. Thereafter, the potential to achieve sufficient local heating resonance inside the target tumor cells (quantified by the absorption cross-section) to be able to destroy them through heat, with no adverse effect on the surrounding normal tissue, will be evaluated. With this in mind, in [20,21] one of the present authors developed a general analytical framework for a graded waveguide structure in which a thin dielectric layer surrounded by some background material is described as a stratified medium, as in Figure 2. The work supports both dispersive and lossy material parameters, and the total scattering matrix can be obtained without the use of mode matching and cascading techniques. With the two media described by a single, effective permittivity function, there is no need for boundary conditions between the media. Consequently, only a single solution of Maxwell’s equations is needed to obtain transmission, reflection and absorption parameters needed to assess the feasibility of the proposed heating method. The graded waveguide description is motivated by its ease of extracting reflection, transmission, and absorption coefficients, relevant for future measurement setups as well as current numerical model setups. Furthermore, there is always a transition region between biological tissues, and no sharp interfaces. The medical application is thus graded by nature.

The general exact analytical fomulas for transmission, reflection, and absorption coefficients in waveguide settings with inserted GNP-substrates are known from our previous work [20,21]. These general fomulas are further developed here to be readily applicable to the calculation of absorption coefficient within the inserted lossy layer only, quantifying the absorption in the GNP-fed cancer tissue. In doing so, a theoretical scaling factor is added as an extension to the theory developed in [20]. The scaling factor is valuable for both the analytical solution and the numerical results when one wants to investigate losses within the thin layer only and eliminate the smaller and unessential absorption in the surrounding lossy medium. This is particularly handy for numerical methods that use scattering parameters, where distinguishing between absorption regions is difficult when the ambient medium is lossy. Although the general analytical results are known, it is of interest to study these analytical solutions in more detail and to compare them to state-of-the-art numerical simulations to confirm their usefulness and prediction power. We therefore compare the numerical simulation results for power transmission, reflection and absorption coefficients, obtained using COMSOL Multiphysics software, with the corresponding results obtained from the modified exact analytic fomulas. The study includes both a simple example of constant complex permittivities, and a more realistic example where a dispersive model of permittivity is used to describe human tissue and the electrophoretic motion of charged gold nanoparticles.

Thus, in this paper, we developed a numerical approach to the graded waveguide model using COMSOL Multiphysics, a modeling software based on the finite element method. The numerical examples in previous work on graded waveguides [20,21] used constant permittivities with respect to frequency, and their magnitudes were chosen with no particular application in mind. In [18,19], material values were chosen based on metal particles immersed in saline water, and with no dispersion. In the present paper, we also include a more realistic numerical example based on dispersive permittivity describing actual human tissue, as well as the dispersive permittivity describing the thin layer with a concentration of GNPs absorbed in such a tissue where their electrophoretic particle acceleration is described by the Drude model.

Due to the similar mathematical notation and descriptions, it is important to particularly emphasize the novelty of the present work, as compared to our previous publications [20,21,22]. In [20], the general theory of TE-wave propagation in the waveguide setting was developed for the first time. It constitutes an important mathematical basis for the present work. However, the work reported in [20] did not include any energy considerations such as, e.g., energy transmission, reflection and absorption, and did not consider the problem of extracting the energy absorption in the GNP-fed thin layer from the absorption in the rest of the infinite waveguide. It does not consider the more realistic finite waveguide either. Furthermore, no particular dispersion models for the GNP-fed target layer and the surrounding tissue were considered, and there were no attempts to numerical verification. These issues are essential for any future simulations and measurements. The work reported in [21] addressed some of these issues. It generalized the results reported in [20] to both transverse electric (TE) and transverse magnetic (TM) polarization, and included a first attempt to provide energy considerations. However, it still does not properly consider the problem of extracting the energy absorption in the GNP-fed thin layer from the absorption in the rest of the realistic finite waveguide, and it does not include any dispersion models for the GNP-fed target layer and the surrounding tissue. In the short conference paper [22], the first attempt to numerically consider the problem of extracting the energy absorption in the GNP-fed thin layer from the absorption in the rest of the infinite waveguide was made. However, in [22], no resolution of the issue of properly considering the problem of extracting the energy absorption in the GNP-fed thin layer from the absorption in the rest of a realistic finite waveguide is provided, and no realistic dispersion models for the GNP-fed target layer and the surrounding tissue are considered. In the present paper, all the above-mentioned issues are finally resolved. Thus, the present paper provides important and novel tools, that are essential for future practical validations and measurements.

Notation and Conventions

The present paper assumes electromagnetic fields oscillating proportional to ${\mathrm{e}}^{\mathrm{i}\omega t}$. This time convention implies that the imaginary part of the relative permittivity ${\epsilon }_r$ must be negative for a passive dielectric material. The vacuum wavenumber is defined by $k=\omega \sqrt{{\mu }_0{\epsilon }_0}$ where ${\mu }_0$ and ${\epsilon }_0$ denotes the permeability and permittivity in vacuum, respectively. The transverse wavenumber is denoted by $k_{\mathrm{t}}$ and longitudinal wavenumber is defined by $k_z=\sqrt{{\omega }^2{\mu }_0{\mu }_r{\epsilon }_0{\epsilon }_r-k_{\mathrm{t}}^2}$. The real and imaginary parts of any complex number $\xi$ are, respectively, denoted by $\Re \left\{\xi \right\}$ and $\Im \left\{\xi \right\}$.

2. Wave Propagation through a Graded Layer

The geometry of the problem is presented in Figure 2. The surrounding non-magnetic lossy waveguide medium is described by the complex relative permittivity ${\epsilon }_{\mathrm{G}}\left(\omega \right)$. The thin lossy non-magnetic layer, inserted about z = 0, is described by the complex relative permittivity ${\epsilon }_{\mathrm{L}}\left(\omega \right)$. In a previous study by one of the present authors [20], it was found that a homogeneous straight waveguide medium with a single thin layer can be described as a stratified medium with frequency-dependent permittivity $\epsilon (\omega ,z)$ being a function of the z-coordinate (waveguide axis direction).

The two different media in the waveguide, as shown in Figure 2, can be modeled as a single stratified medium with a spatially dependent permittivity $\epsilon (\omega ,z)={\epsilon }_0{\epsilon }_r(\omega ,z)$, where ${\epsilon }_r(\omega ,z)$ denotes the relative permittivity, given as a function of the waveguide axis direction

$${\epsilon }_r(\omega ,z)={\epsilon }_{\mathrm{L}}\left(\omega \right)-\left[{\epsilon }_{\mathrm{L}}\left(\omega \right)-{\epsilon }_{\mathrm{G}}\left(\omega \right)\right]{tanh}^2\left(\frac{z}{z_0}\right)$$

(1)

where $2z_0$ represents the thickness of the inserted dielectric layer around the plane z = 0. Far away from the thin dielectric layer ($z\to \pm \infty$) the function (1) behaves as follows

$${tanh}^2\left(\frac{z}{z_0}\right)\to 1\Rightarrow \epsilon (\omega ,\pm \infty )={\epsilon }_0{\epsilon }_{\mathrm{G}}\left(\omega \right),$$

(2)

while at the center position of the thin dielectric layer ($z\to 0$) the relative permittivity (1) becomes

$${tanh}^2\left(\frac{z}{z_0}\right)\to 0\Rightarrow \epsilon (\omega ,0)={\epsilon }_0{\epsilon }_{\mathrm{L}}\left(\omega \right).$$

(3)

This asymptotic behavior ((2) and (3)) is in agreement with the geometry of the problem shown in Figure 2. A geometry with a very thin layer and rapid smooth transition from ${\epsilon }_{\mathrm{G}}\left(\omega \right)$ to ${\epsilon }_{\mathrm{L}}\left(\omega \right)$ and back to ${\epsilon }_{\mathrm{G}}\left(\omega \right)$, is obtained in the limit $z_0\to 0$. The waveguide description presented here is a general theory, where any permittivity model can be employed as long as it satisfies the Kramer-Kronig relations. Two examples of permittivity models [18,19,23] for the two dispersive media, ${\epsilon }_{\mathrm{G}}\left(\omega \right)$ and ${\epsilon }_{\mathrm{L}}\left(\omega \right)$, are given in the next sections of this paper. To describe the wave propagation in a waveguide, without field sources inside ($\rho =0$, $\mathit{J}=0$), we use the following wave equations [20] for the electric and magnetic fields for TE-waves (with $E_z=0$)

$${\nabla }^2\mathit{E}+k^2{\epsilon }_r\left(z\right)\mathit{E}=0,{\nabla }^2{H}_z+k^2{\epsilon }_r\left(z\right)H_z=0.$$

(4)

Here we solve the first of Equations (4) for the electric field $\mathit{E}$, whereby the magnetic field $\mathit{H}$ is readily obtained from the first of Maxwell’s equations, i.e., using

$$\mathit{H}=\frac{\mathrm{i}}{\omega {\mu }_0}\nabla \times \mathit{E}.$$

(5)

Thus, the first of Equations (4) with $E_z=0$ is equivalent to two scalar equations, both of which are of the form

$${\nabla }^2{E}_j+k^2{\epsilon }_r\left(z\right)E_j=0,j\in \{x,y\}.$$

(6)

In the special case of a rectangular waveguide with cross-sectional lengths $d_x$ and $d_y$, we then obtain the overall expressions for the two electric field components in the form

$$\begin{array}{c}\begin{array}{cc}\hfill E_x=& A\tau \frac{n\pi }{d_y}cos\left(\frac{m\pi x}{d_x}\right)sin\left(\frac{n\pi y}{d_y}\right)exp\left(2p\frac{z}{z_0}\right)\hfill \\ & ·{\left[1+exp\left(2\frac{z}{z_0}\right)\right]}^{-2p}{}_2{F}_1\left[a,b,c;\frac{1}{1+exp\left(2z/z_0\right)}\right]\hfill \end{array}\end{array}$$

(7)

$$\begin{array}{c}\begin{array}{cc}\hfill E_y=& -A\tau \frac{m\pi }{d_x}sin\left(\frac{m\pi x}{d_x}\right)cos\left(\frac{n\pi y}{d_y}\right)exp\left(2p\frac{z}{z_0}\right)\hfill \\ & ·{\left[1+exp\left(2\frac{z}{z_0}\right)\right]}^{-2p}{}_2{F}_1\left[a,b,c;\frac{1}{1+exp\left(2z/z_0\right)}\right]\hfill \end{array}\end{array}$$

(8)

where A is a constant proportional to the incident electric field amplitude $E_0$, and $\tau$ is the transmission coefficient which will be given below. Furthermore, ${}_2{F}_1(a,b,c;u)=F(a,b,c;u)$ is the ordinary Gaussian hypergeometric function [24]. In (7) and (8), several short-hand notations are introduced, as also seen in [20,21], and are defined as follows

$$\begin{array}{cc}\hfill a& =2p+\frac{1}{2}+\sqrt{r^2+\frac{1}{4}}\hfill \\ \hfill b& =2p+\frac{1}{2}-\sqrt{r^2+\frac{1}{4}}\hfill \\ \hfill c& =2p+1\hfill \end{array}$$

(9)

with

$$\begin{array}{cc}\hfill {\displaystyle p}& =\mathrm{i}\frac{z_0}{2}\sqrt{k^2{\epsilon }_{\mathrm{G}}\left(\omega \right)-k_{\mathrm{t}}^2}=\frac{\mathrm{i}{k}_{z\mathrm{G}}\left(\omega \right)z_0}{2}\hfill \\ \hfill {\displaystyle r}& =k{z}_0\sqrt{{\epsilon }_{\mathrm{L}}\left(\omega \right)-{\epsilon }_{\mathrm{G}}\left(\omega \right)}\hfill \end{array}$$

(10)

where $k_{z\mathrm{G}}\left(\omega \right)=\sqrt{k^2{\epsilon }_{\mathrm{G}}\left(\omega \right)-k_{\mathrm{t}}^2}$ is the z-component of the wave vector of the asymptotic waves for $z\to \pm \infty$, and the transverse wavenumber $k_{\mathrm{t}}$ is given by

$$k_{\mathrm{t}}^2={\left(\frac{m\pi }{d_x}\right)}^2+{\left(\frac{n\pi }{d_y}\right)}^2$$

(11)

for a rectangular waveguide. Based on the above solutions, from [20], the general expressions for the transmission coefficient ($\tau$) and the reflection coefficient ($\rho$), are

$$\begin{array}{cc}\hfill \tau & =\frac{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }(a+b-c)},\hfill \\ \hfill \rho & =\frac{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }(c-a)\mathrm{\Gamma }(c-b)}\frac{\mathrm{\Gamma }(c-a-b)}{\mathrm{\Gamma }(a+b-c)}.\hfill \end{array}$$

(12)

where $\mathrm{\Gamma }\left(z\right)$ is the Gamma function [24]. In lossless media, the power reflection coefficient is given by $R_P={\left|\rho \right|}^2$ and the power transmission coefficient is given by $T_P={\left|\tau \right|}^2$. These coefficients satisfy $C_{\mathrm{abs}}=1-T_P-R_P=0$, since there is no absorption of energy in the two media.

Power Considerations with Losses

For lossy media, $C_{\mathrm{abs}}=1-T_P-R_P$ is nonzero, such that $C_{\mathrm{abs}}$$(0<C_{\mathrm{abs}}<1)$, quantifies the absorption (electromagnetic energy loss) in the entire waveguide. Plotting this quantity, one can investigate the dependence of the absorption on frequency and on layer size $z_0$. In order to study the absorption in the dielectric layer only, we need to calculate the power reflection coefficient ($R_P$) and power transmission coefficient ($T_P$) over the layer ($-z_0\le z\le +z_0$). In order to exclude the losses in the surrounding medium, we use the following exact analytical fomulas [22]

$$\begin{array}{cc}\hfill T_P& ={\left|\tau \right|}^2·{\left(1+{\mathrm{e}}^2\right)}^{-4\Re \left\{p\right\}}·\chi (a,b,c)\hfill \\ \hfill R_P& ={\left|\rho \right|}^2·{\left(1+{\mathrm{e}}^2\right)}^{-4\Re \left\{p\right\}}·\chi (a,b,c)\hfill \end{array}$$

(13)

where we have defined an auxiliary scale factor $\chi (a,b,c)$ by

$$\chi (a,b,c)={\left|\frac{F(a,b,c;\frac{1}{1+{\mathrm{e}}^2})}{F(c-a,c-b,c-a-b+1;\frac{1}{1+{\mathrm{e}}^2})}\right|}^2$$

(14)

where $F(a,b,c;u)$ is the Gauss hypergeometric function and Equation (10) can be used to relate p to $k_{z\mathrm{G}}{z}_0$. If we want to study the losses, i.e., the absorption, in any finite part of the waveguide, we need to calculate the power reflection coefficient ($R_{Pm}$) and power transmission coefficient ($T_{Pm}$) over some arbitrary layer ($-m·z_0\le z\le +m·z_0$), where m is a number that defines how large is the studied part in the units of $z_0$. It is not necessarily an integer, but can be any fraction. We then obtain the generalized fomulas

$$\begin{array}{cc}\hfill T_{Pm}& ={\left|\tau \right|}^2·{\left(1+{\mathrm{e}}^{2m}\right)}^{-4\Re \left\{p\right\}}·{\chi }_m(a,b,c)\hfill \\ \hfill R_{Pm}& ={\left|\rho \right|}^2·{\left(1+{\mathrm{e}}^{2m}\right)}^{-4\Re \left\{p\right\}}·{\chi }_m(a,b,c)\hfill \end{array}$$

(15)

where we have defined

$${\chi }_m(a,b,c)={\left|\frac{F(a,b,c;\frac{1}{1+{\mathrm{e}}^{2m}})}{F(c-a,c-b,c-a-b+1;\frac{1}{1+{\mathrm{e}}^{2m}})}\right|}^2$$

(16)

When performing the analytical analysis of reflection, transmission and absorption, Equation (14) is sufficient as a scale factor since one is free to assume an infinitely long waveguide. When analyzing waveguides of finite length, which is required by numerical methods and experimental setups, being by necessity of finite length, one can use both (14) and the generalization (16) to relate the losses in the thin layer of thickness $2z_0$ to the losses in a finite waveguide with length $m·2z_0$, which in turn can be seen as a section of an infinite waveguide. From the above fomulas we readily see that the relative scale factor $T_{Pm}/T_{P1}=R_{Pm}/R_{P1}$ is given by

$$\frac{T_{Pm}}{T_{P1}}=\frac{R_{Pm}}{R_{P1}}={\left(\frac{1+{\mathrm{e}}^{2m}}{1+{\mathrm{e}}^2}\right)}^{-4\Re \left\{p\right\}}·\frac{{\chi }_m(a,b,c)}{\chi (a,b,c)}$$

(17)

This is a rather complex scale factor but it is exactly equal for both the power transmission and power reflection coefficients.

3. Dispersive Studies

With regards to the medical application of the graded waveguide, the materials involved are biological tissue as well as coated metal nanoparticles in electrophoretic motion by the applied electric field.

3.1. Biological Tissue

The dielectric properties of biological tissue can be adequately described by the Debye model. In [25], a variant of the Debye equation, the Cole–Cole equation, is employed in an effort to better fit its parameters to experimental measurements of permittivity of various human tissues,

$${\epsilon }_{\mathrm{G}}\left(\omega \right)={\epsilon }_{\infty }+\sum _{n=1}^4\frac{\Delta {\epsilon }_n}{1+{\left(\mathrm{i}\omega {\tau }_n\right)}^{(1-{\alpha }_n)}}+\frac{{\sigma }_i}{\mathrm{i}\omega {\epsilon }_0},$$

(18)

where ${\epsilon }_{\infty }$ is the high frequency permittivity and ${\sigma }_i$ is the static ionic conductivity. In (18), the spectrum of a tissue is described in terms of four Cole–Cole dispersions, where $\Delta \epsilon ={\epsilon }_s-{\epsilon }_{\infty }$ is the magnitude of dispersion with ${\epsilon }_s$ being the static permittivity, $\tau$ is the relaxation time, and the distribution parameter $\alpha$ is a measure of broadening in each dispersion. Each term in essence represents one of four relaxation regions characterized in tissue across a spectrum ranging from Hz to GHz [26]. A library of parameter values for (18) for a large selection of various human tissues can be found in [25,27] as determined by fitting procedures to experimental data [28].

3.2. Electrophoretic Mechanism

A Drude model is assumed to describe the electrophoretic mechanism of GNPs within a host medium. Then, the permittivity function of the thin dielectric layer may be written as

$${\epsilon }_{\mathrm{L}}\left(\omega \right)={\epsilon }_{\mathrm{G}}\left(\omega \right)-\frac{{\omega }_p^2{\tau }_{\mathrm{D}}^2}{1+{\omega }^2{\tau }_{\mathrm{D}}^2}-\mathrm{i}\frac{{\omega }_p^2{\tau }_{\mathrm{D}}^2}{\omega (1+{\omega }^2{\tau }_{\mathrm{D}}^2)},$$

(19)

where ${\epsilon }_{\mathrm{G}}$ is the host medium and ${\omega }_p^2={\sigma }_{\mathrm{D}}/\left({\epsilon }_0{\tau }_{\mathrm{D}}\right)$ is the plasma frequency wherein the Drude parameters may describe an electrophoretic mechanism with static conductivity ${\sigma }_{\mathrm{D}}=\mathcal{N}{q}^2/\beta$ and relaxation time ${\tau }_{\mathrm{D}}=m/\beta$. Here, $\mathcal{N}$ is the number of charged particles per unit volume, q the particle charge, m the mass of the particle, and $\beta$ the friction constant of the host medium, see [11]. The friction constant is given by Stoke’s law $\beta =6\pi {\mu }_{f}a$, with ${\mu }_f$ being the dynamic shear viscosity of the host medium and a the total radius of a GNP consisting of a gold nanoparticle core plus ligands.

4. The Finite Element Model

A three-dimensional model of the waveguide illustrated in Figure 2, with a rectangular cross-section, was developed in COMSOL Multiphysics. Material properties were described by the spatial function (1) with complex permittivities ${\epsilon }_{\mathrm{L}}$ and ${\epsilon }_{\mathrm{G}}$. The waveguide walls acted as perfect electric conductors. Rectangular ports at either end of the waveguide excite and absorb the propagating modes within, and were backed by perfectly matched layers. The waveguide was given a finite length sufficiently larger than $z_0$, so that the ports were located far away from the layer, in the limit (2). The reflection ($R_{Pm}$) and transmission ($T_{Pm}$) power coefficients were calculated using the scattering parameters obtained from COMSOL at the reflection ($S_{11}$) and transmission ($S_{21}$) ports, which were then multiplied by the scale factor (17) to account for the ambient losses $\Im \left\{{\epsilon }_G\right\}$, where the number m describes the ratio of waveguide length (i.e., distance between the two ports) to thin layer length $2z_0$.

The simulations presented in this paper were run on a computer equipped with an Intel i7-10700K processor with 64 GB RAM, running on Windows 10. In the case of a waveguide length of 17 cm, the solution time was 1 s for $f=1$ GHz and 16 s for $f=3.5$ GHz.

5. Results

We demonstrate the computational model described in Section 4, and make a comparison with the exact analytical Equations (13), for the case of a rectangular waveguide with dimensions $6\times 3$ cm, but this can be re-scaled to study other frequency ranges, and changing the waveguide cross-section can be achieved with minor adjustments. The propagating wave is the lowest TE mode, TE${}_{10}$. The results are presented as the power reflection $R_P$, power transmission $T_P$, and power absorption $C_{\mathrm{abs}}$ over the thin dielectric layer of width $2z_0$. First, the model is tested for constant and complex permittivites, similar to those found in [21] for TM${}_{11}$. Then, frequency-dependent complex permittivites are demonstrated using the dielectric functions described in Section 3.

5.1. Non-Dispersive Example

The power coefficients over the thin dielectric layer for constant complex permittivities ${\epsilon }_{\mathrm{L}}=4-0.8\mathrm{i}$ and ${\epsilon }_{\mathrm{G}}=2-0.1\mathrm{i}$ are shown in Figure 3. In Figure 3a the transmission, reflection and absorption are shown as functions of the operating frequency f with fixed layer width $z_0=1\mathrm{cm}$. The relative error difference averages around $0.02\%$. Figure 3b shows the coefficients as functions of the layer width $z_0$, for a selection of fixed frequencies, where the relative error averages at $1.0\%$. The cutoff frequency for both cases is $f_c=1.77$ GHz.

5.2. Dispersive Example

The materials are here described by the dielectric models in Section 3, although in general any dielectric model or dispersive data table may be used. For this demonstration, a healthy breast fat tissue is simulated in the waveguide. The GNPs are localized within the layer of width $2z_0$, where the concentration of particles gradually increases toward the center $z=0$. The background material ${\epsilon }_{\mathrm{G}}\left(\omega \right)$ is given by the four-term Cole–Cole Equation (18) with parameters describing breast fat as in [27], while the thin dielectric layer ${\epsilon }_{\mathrm{L}}\left(\omega \right)$ is given by the Drude function (19) with breast fat as its host medium.

The parameters for the Drude function (19) were selected based on the parameter study in [18], with some modification to account for a different host medium. The radius of GNPs including ligands were $a=2.5$ nm with gold core radius $a_{\mathrm{Au}}=0.75$ nm. The mass of a GNP is then $m=(4\pi a_{\mathrm{Au}}^3/3){\rho }_{\mathrm{Au}}+[4\pi (a^3-a_{\mathrm{Au}}^3)/3]{\rho }_{\mathrm{L}}$, where ${\rho }_{\mathrm{Au}}=19,300$ kg m${}^{-3}$ and ${\rho }_{\mathrm{L}}=1000$ kg m${}^{-3}$ are the mass densities assumed for gold and ligands, respectively. The net charge of GNPs is modeled as $q=(3a_{\mathrm{Au}}+0.5a_{\mathrm{Au}}^2)e_0+n_{\mathrm{L}}{e}_0$, as in [11,18], where $e_0$ is the electron charge and $n_{\mathrm{L}}=250$ is in this case the chosen net electron count in the ligand shell. The particle density has been calculated as $\mathcal{N}=\varphi /(4\pi a^3/3)$ where $\varphi =2.5·{10}^{-2}$ denotes the volume fraction of GNPs within the layer. The dynamic viscosity of the adipose host medium is assumed to be ${\mu }_f=3·{10}^{-2}$ Nsm${}^{-2}$.

With the above permittivites for the thin layer and surrounding medium, the ${\mathrm{TE}}_{10}$ response of the graded waveguide with layer width $z_0=0.5$ cm can be found in Figure 4, plotted for frequencies above the cutoff frequency $f_c=1.08$ GHz. The relative error between analytical solution and computational results is under $0.1\%$. Figure 5 illustrates the data processing accomplished to obtain the simulation results found in Figure 4. The output from COMSOL is given in S-parameters measured at the ports at either end of the waveguide, i.e., power reflection and transmission throughout the entire waveguide of 17 cm in length. Since we are only interested in the power coefficients within the central layer, the S-parameters are multiplied by the scaling factor (17) plotted in Figure 5b.

6. Discussion

From the results presented in Figure 3 and Figure 4, we see that there is an excellent agreement between the numerical results, obtained using the COMSOL simulation software, and the analytical results obtained using (13).

In the dispersive example above, we emphasize that the electrophoretic Drude description (19) involves several design parameters, e.g., the molecular composition of ligands affecting their electron count $n_{\mathrm{L}}$, and it is left for future work to investigate the feasibility of the medical application in regards to these parameters. Although the material properties chosen in the dispersive study are realistic themselves, their combination do not necessarily represent the description of the medical application. A step in this direction could be to replace ${\epsilon }_{\mathrm{G}}$ to describe a malignant breast tissue instead of normal breast tissue. The difference in microwave properties between the two have shown that malignant tumor is estimated at almost 5–10 times larger [29], which will affect its response to electromagnetic radiation. Additionally, with cancerous tissue having significantly higher water content than normal tissue, and adipose tissue such as breast fat having reduced water content, the motion of GNPs in their host medium will be affected via the friction constant $\beta$. As such, both Drude parameters and the layered composition of involved materials will be tuned further in order to achieve specific goals such as maximum absorption resonances for the medical application, which will be the objective of our continued efforts.

Compared to many other tissues in the human body, breast fat and other adipose tissues is less absorbing due to lower water content, where the imaginary part of the permittivity has values around 0.8–1.5 in the low GHz range. Even so, over the entire waveguide (17 cm in the dispersive example) at most $3\%$ is reflected and a fourth of that is transmitted via the thin layer, as seen in Figure 5a. The remainder is absorbed throughout the surrounding tissue as well as the thin layer. Comparing this to the result in Figure 4 it is clear that the relative scale factor (17) is valuable in extracting meaningful information on the response within the thin layer only.

7. Conclusions

We studied absorption phenomena in a hollow waveguide with an inserted graded dielectric layer for the case of TE-wave propagation. The proposed waveguide model is developed to be used in a study of a potential cancer treatment by heating of GNPs inside the cancer cells. A scaling factor was introduced to the exact analytical fomulas for transmission and reflection coefficients, allowing the study of absorption only within the central dielectric layer of the waveguide, the region most relevant for the intended medical application. The scaling factor was further generalized to handle losses in a layer of arbitrary thickness. Thus, it can be used to study absorption in a finite waveguide, as is required by experimental setups and many numerical methods, being by necessity of finite length. The fomulas were validated in accordance with a numerical graded waveguide model developed in a commercial finite element software. The analytical solutions and the finite element model were compared for both non-dispersive and dispersive materials and found to be in excellent agreement. In the latter case, dispersion models describing GNPs in electrophoretic motion submerged in human tissue were used for the thin dielectric layer, surrounded by lossy human tissue. The S-parameter results were applied with a relative scaling factor as a function of the waveguide length, thin layer thickness, and material permittivites, and successfully demonstrated the usefulness of the scaling factor in distinguishing between absorption within the thin layer and the lossy ambient. Thus, we conclude that the proposed model provides a valuable analytical and numerical tool for future research of RF absorption phenomena in GNP-fed cancer tissue.