## 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 \({}^{\circ}\)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 [PERSON]'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.