Heating Power of Millimeter-Sized Implanted Coils for Tumor Ablation: Numerical-Analytic Analysis and Optimization

Minimally invasive thermal ablation procedures of tumors with implanted devices are very promising, especially for the repetitive treatment of deep-seated tumors. The implanted devices are heated without contact by an alternating magnetic field from outside the patient’s body. In this paper, the heating power of millimeter-sized implanted coils is analyzed and optimized with a numerical-analytic analysis and the dependencies on spatial, electrical, and magnetic parameters are evaluated and presented for being able to choose the optimum implanted coil for a specific set of parameters. The analysis is done with focus on the implanted coils based on a homogeneous alternating magnetic field. A heating power of 1.5 W required for achieving an adequate rise of tissue temperature is determined in a thermal analysis and the corresponding specific absorption rate (SAR) is evaluated along with the power transfer efficiency (PTE) and the coupling coefficient for different types of implanted coils. For uncompensated implanted coils, a SAR of 306 mW/kg, a PTE of 4.62 · 10-3 and a coupling coefficient of 2.49 · 10-3 is achieved by a magnetic field strength of 1727 A/m, whereas a SAR of 1.84 mW/kg, a PTE of 436 · 10-3 and a coupling coefficient of 2.3 · 10-3 is achieved by a magnetic field strength of 134 A/m for serial compensated implanted coils. With this, the ratio of heating power to required magnetic field strength is maximized, which reduces the risk of unwanted heating of healthy tissue and other implanted devices and therefore enhances the safety as well as the well-being of the patients.


I. INTRODUCTION
C ONTACTLESS energy transfer (CET) is very advantageous in various fields of applications, such as electrical machines and dynamic charging of electric vehicles [1]- [9] and supplying energy to and monitoring of implanted medical microsystems [10]- [12]. A very important application of CET is the non invasive and the minimally invasive treatment of cancer by heating up and by ablating tumor tissue (hyperthermia). Here, tumor tissue is heated directly by eddy currents resulting from radiofrequency magnetic fields or by devices implanted or injected into the tumor and heated by an alternating magnetic field based on magnetic or ohmic losses, such as magnetic nanoparticles and permanently implanted devices [13]- [16].
Generally, hyperthermia by permanently implanted devices can be divided into two categories: Heat generation by applying electrical currents directly to the tumor tissue with electrodes [17] and heating tumor tissue indirectly by generating heat in an implanted device, which is referred to as magnetically mediated hyperthermia (MMH) [18], [19]. Since the electrical properties of human tissue change with tissue temperature and differ with tissue type, the generation of heat by electrical currents (eddy currents or currents applied directly to the tissue by an implanted device) strongly depends on tissue temperature and type of tissue. Hence, an optimization of power absorption is difficult to achieve.
Schematic representation of m single secondary coils, which are penetrated by a magnetic field #» HP created by a single or multiple primary coils. The magnetic flux ΦPm, which is received by Lm from the primary coil or coils, and the magnetic flux Φ12, which is created by L1 and received by L2, are presented along with the magnetic conductivities Λ12, Λ1m, and Λ2m between all single coils. Λ12, Λ1m, and Λ2m are equal to Λ21, Λm1, and Λm2, respectively.

Coil
Body Tissue In contrast to that, power absorption can be optimized significantly with MMH by choosing the appropriate implant design and material properties due to power absorption is independent of tissue properties. Additionally, a more localized heating of tissue can be accomplished with MMH. The optimization of power absorption and the localization of tissue heating are essential for the well-being of the patients as unwanted heating and influencing of healthy tissue are minimized.
In the last decades, intensive research has been done on the feasibility and on heat generation with respect to a contactless thermal treatment of tumors with inductively heated implanted devices [20]- [27]. Different models for wire wound coils were presented by the authors in [28]- [33] based on numerical and analytic approaches. The frequency and the electrical conductivity for an implanted coil were analyzed and optimized in [34], whereas the radii of the primary and the secondary coils and the radii of the coil wires as well as the frequency were optimized for maximum heating efficiency in [35] for MMH. Different inductive links and coil type implants were investigated considering reactive power compensation with respect to the Q-factor [29], with respect to the maximum deliverable power under a specific absorption rate (SAR) constraint in the lower MHz range [36], and with respect to a secondary coil figure of merit comprising Qfactor, coupling coefficient and secondary coil efficiency in the middle to upper MHz range with focus on the secondary coil [37]. A maximum power absorption with respect to an infinitely long circular cylinder is presented in [38]. In [39] and [40], an optimization of the power absorption per unit volume for ferromagnetic implanted devices heated by eddy currents is performed based on the optimum induction number by replacing a single solid ferromagnetic implant with multiple strands of wire fitting to the same cross sectional area. The authors in [41] analyzed the power absorption of a ferrite core with high permeability surrounded by a metallic sheath. The power absorption was measured and calculated based on the measured effective relative permeability of the ferrite core.
In the present paper, the heating power of wire wound and foil wound implanted coils, which corresponds to the power delivered to load (PDL), is analyzed by numerical and analytic calculations. As shown in Fig. 2, these coils are positioned within the tumor by creating a small access to the body tissue and inserting a narrow tube (trocar), through which the coils are transported to the tumor. This is referred to as minimally invasive operation technique. The resulting size restrictions with respect to the implanted coils are taken into account in the analysis done in this publication. The dependency on the diameter of the coil wire and the thickness of the foil, respectively, as well as on the electrical resistivity of the conductor material, on the frequency of the alternating magnetic field, and on the permeability of the coil core is determined and an optimization for achieving maximum heating power with respect to the spatial restrictions of a minimally invasive operation technique is done. The influence of the dimensions and of the magnetic properties of the implanted secondary coil's core on the spatial distribution of the magnetic field and the influence of the dimensions, the electrical properties and the spatial position of the coil windings on the resulting heating power are determined precisely. This has not been done in scientific literature yet. Moreover, a higher heating power per unit length is presented in this publication compared to the results reported in existing scientific literature.
The analysis in this publication focuses on achieving maximum heating power for a given alternating magnetic field. This enables to optimize the implanted coils and the field generating coils separately by assuming a very low coupling between the primary coils and the secondary implanted coil and therefore assuming a current with a constant magnitude flowing in the primary coils. Based on the numerical calculations with the finite element method (FEM), the inductance and the magnetic flux within the implanted secondary coil are evaluated precisely by determining the spatial distribution 2 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ of the magnetic field influenced by the ferromagnetic core of the coil and by determining the coupling between every single winding of the implanted coil. Based on this, the heating power is calculated analytically. A thermal evaluation is done to evaluate the minimum required heating power for achieving an adequate rise of tissue temperature with respect to conduct a thermal tumor ablation. The resulting SAR, the resulting power transfer efficiency (PTE), and the coupling coefficient are determined based on a model of the human body and based on specific primary coil configurations for the worst case scenario of deep-seated tumors, which have been analyzed by the authors in [42].
In Section II of this publication, the basics of magnetic coupling are presented, followed by the settings and the structure of the numerical-analytic analysis. In Section IV and Section V, the numerical and the analytic analyses are described in detail. Subsequently, the results of the numericalanalytic analysis are presented in Section VI along with the results of the validation, followed by a thermal analysis and the evaluation of the SAR, of the PTE, and of the coupling coefficient. Finally, the results are discussed in Section VIII and some concluding words are given in Section IX.

II. MAGNETIC COUPLING BASICS
The magnetic field created by the single windings of a secondary coil and the magnetic field created by a primary coil for CET are inhomogeneous within the ferromagnetic core of the coil. Hence, for precisely evaluating the inductance and the induced voltage of a wire wound or a foil wound coil with multiple windings and a ferromagnetic core, the coupling between every single winding and the magnetic flux created by the primary alternating magnetic field in every single winding has to be determined. In this section, the basics of magnetic coupling required for calculating the inductance, the induced voltage and the resulting current of a secondary coil for an arbitrary primary magnetic field are presented.

A. MAGNETIC CONDUCTIVITY
The inductance of a coil and the coupling between coils can be expressed in terms of the magnetic conductivity Λ, which is a measure for the influence of the material properties along the field lines of a magnetic field and for the spatial properties of a single coil or a coupled coil configuration. According to Ampère's Law the following equation can be derived where H s denotes the average magnitude of the magnetic field strength along the average magnetic field line with the length l and I denotes the current, which is applied to a current loop with N windings [43]. Thus, the magnetic flux Φ reveals to where A denotes the area covered by the current loop and B A denotes the average magnetic flux density, which is perpendicular to A. Therefore, according to (3), the magnetic conductivity Λ of a single current loop (N = 1) can be determined with by evaluating the magnetic flux Φ, which is created by applying a current I to the current loop (measurement, analytic or numerical calculation). The inductance L of a coil consisting of one or multiple windings located close to each other formed by this current loop reveals to where Ψ denotes the linked magnetic flux [44].

B. SECONDARY COIL
In Fig 1, a schematic representation of m single secondary coils and the corresponding magnetic conductivities between the coils is shown along with the exemplary magnetic fluxes Φ Pm and Φ 12 . Φ Pm denotes the magnetic flux, which is created by the primary coil or coils and which is received by L m , and Φ 12 denotes the magnetic flux, which is created by L 1 and which is received by L 2 . The overall magnetic flux Φ k of the kth single coil is determined by summing up all single magnetic fluxes Φ nk generated by all coils and received by the kth coil. Φ k results in where m = N S denotes the total number of secondary single coils, N n denotes the number of windings of each single secondary coil, and I n denotes the current in each single secondary coil. With N 1 = N 2 = · · · = N m = N sc (all single coils have N sc windings) and I 1 = I 2 = · · · = I m = I S (the same current I S flows in all single coils), the voltage of each single secondary coil reveals to where ω denotes the angular frequency of the primary magnetic field [44]. In case all single secondary coils are VOLUME 4, 2016 3 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.  connected in series to form a combined secondary coil, the voltage U S of this combined secondary coil results in Thus, the voltage U S , which is created by the alternating magnetic field in the combined secondary coil with the inductance L S , reveals to In case a current with a constant magnitude is supplied to the primary coils or the coupling between the primary and the combined secondary coil is very low, Ψ PS does not depend on the current I S in the combined secondary coil. This is discussed in detail in Section III-C. With where R S denotes the resistor, which is connected to the combined secondary coil and which generates heat, this leads to the electrical equivalent circuit of the secondary coil shown in Fig. 3a for the settings of the analysis carried out in this paper. These settings are introduced in Section III. In this publication, R S is assumed to be the resistance of the coil conductor and therefore belongs to the combined secondary coil (see Section III).

III. ANALYSIS SETTINGS
The amount of energy, which causes a rise of temperature in the implanted secondary coil, depends on various parameters, such as the magnetic field strength and the spatial, magnetic and electrical properties of the implanted coil as well as on the type of reactive power compensation. This section presents the settings of the analysis, how the different parameters are taken into account, and how the results are evaluated efficiently.

A. DIMENSIONS
For implanting the secondary coil into the tumor, a minimally invasive operation technique is used in the contactless thermal tumor ablation procedure on which this paper is based on. Hence, the diameter of the implanted coil must not exceed the dimension of the working channel. Additionally, the length of the coil is restricted for not getting stuck in bent working channels. According to that, a maximum coil diameter of d max = 1.5 mm and a maximum coil length of l max = 20 mm is assumed. Therefore, the diameter of the coil core has to be decreased when increasing the diameter of the coil conductor and vice versa as the overall dimensions of the implanted coil has to be at maximum for maximizing Ψ PS . Increasing the diameter of the implanted coil increases Ψ PS due to an increased cross sectional area and, for a coil core with a relative permeability of µ r > 1, to an increased magnetic flux density. Furthermore, increasing the length of the implanted coil increases the magnetic flux density for a coil core with a relative permeability of µ r > 1 as well. Additionally, depending on the type of conductor, more windings can be added to the coil.

B. COIL TYPES
Two different types of cylindrical implanted coils are analyzed in this work. All coil types comprise a cylindrical core consisting of a material which has a relative permeability of µ r ≥ 300 and which is electrically non-conductive. The relative permeability of the core is assumed to be independent of frequency and magnitude of the alternating magnetic field. The conductor is assumed to have a relative permeability of µ r = 1 (e.g. copper) and the electrical resistivity ρ c . These coil types are selected for the analysis as the dimensions of the coils can be adapted to use the maximum space provided by the minimally invasive operation technique based on the cylindrical coil shape.
The resistor R S shown in the electrical equivalent circuit (see Fig. 3a), which heats up the implanted coil, is assumed to be the resulting resistance of the coil conductor. This leads to an uniform rise of temperature along the implanted coil instead of single hot spots, which increases the volume of heated tumor tissue and is advantageous for a uniform heat distribution inside the tumor. Thus, R S depends on the conductor material, on the number of windings, and on the cross sectional area of the conductor. Additionally, the frequency of the alternating magnetic field influences R S .
The following coil types are analyzed in this work: • Wire wound coil (WWC): The conductor of this coil type is a wire with the diameter d W , which is wound around the cylindrical core. The number of windings depends on the diameter of the wire. Hence, a decreased diameter of the wire leads to an increased number of windings and therefore to an increased magnetic flux Ψ PS , which is received from the primary coils, and an increased conductor resistance. For the WWC, solely wire diameters, which result in an integer number of windings, are taken into account. A schematic representation of the WWC is shown in Fig. 4a.
• Foil wound coil (FWC): A conductive foil with a thickness d F is wound around the cylindrical core. The overall thickness of all windings is referred to as the thickness of the conductive layer d L . In contrast to the WWC, the number of windings and the diameter of the core can be chosen independently by adjusting d F . Furthermore, the coupling between the single windings is expected to be higher than between the single windings of a WWC, which leads to an increased inductance for the same number of windings. A schematic representation of the FWC is shown in Fig. 4b.
For simplifying the analysis, the single windings of the WWC are taken into account by single current loops connected in series instead of considering a helix. Additionally, the conductors are assumed to have a insulation layer on the outside boundaries to realize the insulation between the single windings. As this layer usually is very thin, it is not taken into account in the dimensions of the numerical and analytic model of the coils. Additionally, the parasitic capacitance of the secondary coils is neglected due to a low maximum frequency of the primary alternating magnetic field.

C. PRIMARY SIDE ALTERNATING MAGNETIC FIELD
Deep-seated tumors are the worst case scenario and therefore the most challenging situation for a contactless transfer of heating energy to the tumors as the energy has to be transferred over a certain distance (more than 25 cm in case a body diameter of 50 cm is assumed) from outside of the patient to the implanted device in the tumor. Hence, the primary coils for generating the alternating magnetic field have to be considerably large (approximately 50 cm in diameter, depending on primary coil system) compared to the implanted secondary coil inside the tumor (1.5 mm in diameter) due to the maximum diameter of the implanted secondary coil is limited by the minimally invasive operation technique used for positioning the secondary coil in the tumor. According to this, the coupling between the primary coils and the secondary coil is extremely low.
In the implanted secondary coil, heat is generated by an alternating magnetic field. This magnetic field is generated by one or multiple primary coils, which are located in a certain distance to the implanted secondary coil. Due to this analysis is based on the worst case scenario of deep-seated tumors, the distance between the implanted secondary coil and the primary coils is assumed to be large compared to the dimensions of the secondary coil. Therefore, the magnetic field in the surrounding area of the secondary coil is assumed to be homogeneous. Furthermore, only the component of the magnetic field parallel to the axis of the implanted coil is taken into account as this component solely contributes to heat generation. The contribution to heat generation of the remaining components, which consequently are perpendicular to the secondary coil axis, can be neglected as these components do not contribute to voltage induction in the secondary coil and no eddy currents are created in the coil core due to the core is assumed to be electrically nonconductive. Additionally, all values for the magnetic field strength are considered to be peak values.
Generally, the current I P in the primary coil (in case of a single primary coil) can be expressed as where U P denotes the voltage supplied to the primary coil, Λ P denotes the magnetic conductivity of the primary coil and N P denotes the number of windings of the primary coil. The magnetic conductivity between primary and secondary coil is described by Λ PS . The magnetic flux Ψ PS , which is created by the primary coil and received by the implanted secondary coil, can be expressed as The voltage of the secondary coil can generally be expressed as (13) Thus, due to Λ PS ≪ Λ P and Λ PS ≪ 1 H for the spatial and magnetic properties, on which this analysis is based on, the magnitude of the primary current I P , the magnitude of the magnetic flux Ψ PS , and the magnitude of the voltage of the secondary coil U S are considered to be independent of the current I S in the secondary coil. Hence, according to (11) and (12), a current source with a constant magnitude is assumed to be used in this publication for supplying energy to the primary coils. With respect to maximizing the heating power, this represents the worst case scenario as the heating power increases in case of a higher coupling is assumed. Furthermore, due to Λ PS ≪ Λ P , Λ PS ≪ 1 H, and Λ S ≪ Λ P and according to (13), the influence of the properties of the primary coils on the voltage U S of the secondary coil as well as the influence of the current I S on the current in the primary coils can be neglected. Therefore, the number, the type and the construction of the primary coils has not to be taken into account for the analysis carried out in this paper. The evaluation of the heating power P S,H can be done based on a given spatial distribution of the magnetic field in the area surrounding the tumor. Thus, the analysis carried out in this paper is valid for any primary coil or coil configuration supplied by an alternating current source with a constant magnitude. For

D. REACTIVE POWER COMPENSATION
For increasing the heating power in the secondary coil, the reactive power in the secondary coil has to be compensated. Two basic compensating strategies are commonly used for this, the serial compensation and the parallel compensation, in which a compensating capacitor C S is connected in series to the heating resistor R S or in parallel to the heating resistor R S . As the heating resistor in this analysis is represented by the conductor resistance of the secondary coil, solely a serial reactive power compensation can be realized. The electrical equivalent circuit of the secondary coil with serial reactive power compensation is shown in Fig. 3b. According to this, the analysis in this publication is done for uncompensated (UC) secondary coils and for secondary coils with serial reactive power compensation (SC).

E. STRUCTURE OF THE ANALYSIS
The analysis carried out in this work is based on a combination of numerical and analytic calculations. By a numerical calculation with FEM, the influence of the secondary coil core on a homogeneous magnetic field, which represents the magnetic field created by the primary coils, and on the magnetic fields created by the single coil windings is determined. The spatial properties as well as the magnetic properties of the implanted coil are taken into account in the numerical calculation. Based on the resulting magnetic fields, the magnetic conductivity Λ S of the implanted coil and the magnetic flux Ψ PS , which is created by the primary coils and received by the implanted secondary coil, are evaluated. The numerical calculation is described in detail in Section IV. Subsequently, with Λ S and Ψ PS , the power P S,H , which heats the secondary coil, is evaluated analytically with respect to the spatial and electrical properties of the implanted coil as well as by taking into account the magnetic field strength and the frequency of the primary magnetic field and the type of reactive power compensation. The analytic calculation is described in detail in Section V. The structure of the analysis is presented in Fig. 5. Gray blocks represent the calculations and white blocks represent the input data for these calculations. The output of the single calculation steps is shown on the arrows, which indicate the flow of data. The structure of this analysis enables a fast and efficient evaluation of the heating ability of an implanted coil. As the influence of a cylindrical core with a relative permeability of µ r > 1 on the spatial distribution of the magnetic field can only be expressed as a function of elliptic integrals, which generally cannot be expressed in terms of elementary functions, this is done by a FEM simulation, which is, especially for a high number of coil windings, time consuming due to the magnetic field has to be evaluated at various coordinates for calculating Λ S and Ψ PS . Generally, the entire analysis could be done solely by a FEM simulation, which would create high computational costs due to different electrical parameters have to be taken into account and due to the FEM simulation has to be recalculated for every single value for all electrical parameters. By considering the electrical parameters and the type of reactive power compensation in an analytic way based on the results of the FEM simulation, the flexibility of the entire analysis is enhanced and the duration is reduced.

IV. NUMERICAL ANALYSIS
The numerical calculations are controlled by MATLAB and are done with COMSOL based on a FEM model by using the LiveLink interface of COMSOL and MATLAB. For reducing computational costs without loosing accuracy, it is taken 6 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and  10 100 a Electrical resistivity of silver advantage of the rotational symmetry of the coil types in the FEM model. Therefore, solely the values in the rz-plane at φ = 0 are calculated and rotated around the middle axis of the core for evaluating Λ S and Ψ PS . The FEM models of a WWC and a FWC are shown in Fig. 6 along with the magnetic field lines originating from a homogeneous primary magnetic field. In the numerical analysis, parameter sets for the coil wire diameter d W (WWC), for the thickness of layer d L (FWC), for the number of windings N S (FWC, N S for WWC results from d W ), and for the relative permeability µ r of the core (WWC and FWC) are calculated. The results of the numerical analysis are summarized and presented in Fig. 7 and an overview on the parameter sets used in the numerical and the analytic analysis is given in Table 1.

A. MAGNETIC CONDUCTIVITY OF THE SECONDARY COIL
Due to the magnetic conductivity between two coils or two windings does not depend on direction (e.g. Λ 12 = Λ 21 ), only half of all winding combinations have to be calculated, according to which is derived from (8).
The coupling and hence the magnetic conductivity Λ nk between the single windings is different for every combination of two windings. Every single winding therefore contributes differently to the combined magnetic conductivity Λ S of the secondary coil. Λ nk depends on the position relative to the core and the distance between the windings, on the diameter of the winding as well as on the relative permeability µ r of the core.
For calculating Λ nk for all winding pairs, a current I FEM is applied successively to every winding and the resulting magnetic flux Φ nk , which is received from the other windings and the source winding itself, is evaluated. In this evaluation step, the winding, to which the current is applied to, is the only source of a magnetic field. No magnetic field is generated by the primary coils in this step. According to (4), Λ nk is calculated with and N = 1. Subsequently, Λ S is calculated with (14).

B. RECEIVED MAGNETIC FLUX FROM PRIMARY COILS
As described in Section III-C, a homogeneous alternating magnetic field parallel to the axis of the core is assumed to be created by the primary coils in the area surrounding the tumor. This magnetic field is implemented as boundary condition on the outside boundaries of the FEM model with a magnitude of H FEM = 1 A m . Furthermore, no current is applied to any winding of the secondary coil in this evaluation step. Due to Ψ PS ∝ H FEM , Ψ PS can be calculated analytically for any magnitude of the magnetic field with the result of this numerical calculation.
The magnetic flux Φ Pk , which is received from the primary coils, is different for every single winding of the implanted secondary coil and depends on the position of the winding relative to the core, on the diameter of the windings, and on the relative permeability of the core as well as on the local magnitude of the magnetic flux density. Φ Pk is determined for every single winding and subsequently the linked magnetic flux Ψ PS is evaluated according to (8) with As a single winding represents a coil with one winding, N sc results in 1 for the analysis carried out in this paper.

V. ANALYTIC ANALYSIS
Based on the results of the numerical analysis, the heating power P S,H , which is generated in the secondary coil, is evaluated analytically with MATLAB. In contrast to the Conductor resistance RS including the resistance of the short circuit connection from the first to the last winding of the coils and the resistance, for which the maximum heating power is achieved, of WWC and FWC with 10 windings for the diameter of the wire (WWC) and for the thickness of the conductor layer (FWC). f = 100 kHz and ρc = 17.25 nΩ · m is assumed to represent the electrical resistivity of copper according to [45].
numerical analysis, the electrical properties of the secondary coil and the type of reactive power compensation are taken into consideration, whereas the same parameter sets for the spatial parameters are used for both analyses. Additionally, the magnitude of the magnetic field strength and the frequency of the primary alternating magnetic field are taken into account analytically. By applying different input values for Λ S and Ψ PS and by applying different equations for determining the conductor resistance R S of the secondary coil, the same analytic model can be used for WWC and FWC.

A. SECONDARY CONDUCTOR RESISTANCE
The conductor resistance of the secondary coil R S is calculated with respect to the electrical resistivity ρ c of the conductor material, the dimensions of the implanted coil, and the construction and number of windings. For this, the bending of the conductor is taken into account, which leads to a lower resistance on the inner side of the conductor (lower diameter) compared to the outer side of the conductor (higher diameter). The resistance of the short circuit connection R SC from the first to the last winding of the coils is taken into account by assuming a thin copper foil underneath the wire windings (WWC) with a resistance of R SC = 500 mΩ and by assuming two thin copper wires on each side of the coil with an overall resistance of R SC = 1 mΩ (FWC). The dimensions of the short circuit connection are neglected when determining the diameter of the core, the diameter of the coil wire d W , and the thickness of the layer d L based on the maximum coil diameter d max and the maximum coil length l max (see Fig. 4). Additionally, for serial compensated secondary coils, an equivalent series resistance (ESR) of R ESR = 2 mΩ is assumed for the compensating capacitors. For both types of secondary coils, the conductor resistance reveals to In Fig. 8, R S for WWC and FWC with 10 windings is shown assuming an electrical resistivity ρ c = 17.25 nΩ · m for representing a copper conductor according to [45].
With increasing frequency of the alternating magnetic field, the resistance of the secondary coil is influenced by the skin effect. This is considered in the analytic model by determining the skin depth and adjusting the cross sectional area of the conductor accordingly in case the diameter of the wire (WWC) or the thickness of the layer (FWC) exceeds the skin depth. The influence of the proximity effect on the conductor resistance is neglected in this analysis.

B. HEATING POWER
The temperature of the implanted secondary coil is increased based on the heating power P S,H . This is one of the key parameters for generating an appropriate rise of temperature in the surrounding tumor tissue and hence for conducting a successful ablation of the tumor. The heating power is converted into heat by the resistance R S of the coil conductor. With the power S S in the secondary coil reveals to The power contributing to heat generation is determined with and maximized by choosing the appropriate compensating capacitor C S for the serial reactive power compensation. With this, the heating power in the secondary coil reveals to In case of an uncompensated secondary coil, the maximum possible heating power P S,H,max reveals to for a conductor resistance whereas the heating power P S,H increases with decreasing conductor resistance R S for a serial compensated secondary  coil according to (21). Hence, the electrical resistivity ρ c of the coil conductor material has to be as low as possible and a coil design, which minimizes R S and which maximizes |Ψ PS |, has to be chosen for achieving maximum heating power. R S,max for both secondary coil types resulting from a copper conductor is shown in Fig. 8.

VI. RESULTS AND VALIDATION
To validate the combined numerical-analytic analysis, FEM simulations are carried out for specific parameter sets and the resulting heating power of both analyses are compared.
No short circuit connection resistances between the start and the end of the coil conductors are taken into account for the validation with FEM. In Fig. 9, the comparison for WWC and FWC with a low conductive conductor material is shown for one set of parameters. Additionally, the results for a very high conductive exemplary conductor material are compared for validating the consideration of the influence of the skin in the analytic model. For this, an electrical resistivity below the parameter set used in this paper is assumed for increasing the influence of the skin effect. The comparison shows a good correspondence of the numerical-analytic results to the 10 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.   results of the FEM simulation for the parameter sets applied in Fig. 9 as well as for further parameter sets with different frequency and different relative permeability of the core. In addition to the validation by FEM simulations, a basic experimental prototype measurement is done. For this, a nearly homogeneous alternating magnetic field with H P = 1840 A m and a frequency of 102 kHz is generated by two primary coils in a Helmholtz configuration. A single winding of copper foil with a thickness of 40 µm is applied to a cylindrical ferrite core with a diameter of approximately 1.45 mm, a length of 20 mm, and a relative permeability of approximately 2300 for representing an uncompensated FWC with a single winding. With this, the heating power results in approximately 0.7 W, which corresponds to the heating power of 0.67 W resulting from the numericalanalytic calculations. However, for a conclusive experimental prototype measurement, a precise measurement setup has to be realized subsequent to this paper along with different types of implanted coils.
In case the resistance of the short circuit connection and the ESR is low compared to the resistance of the coil conductor and as long as the thickness of the conductive layer d L remains constant and the skin effect can be neglected, the number of windings of FWC does not influence the resulting heating power. However, in case the short circuit resistance, the ESR, and the influence of the skin effect has to be taken into account, the maximum heating power increases with increasing number of windings due to |Ψ PS | RS is increased by reducing the thickness of each single winding, which decreases the influence of the skin effect, and by increasing RS RSC+RESR . As shown in Fig. 10, WWC and FWC reveal the same values for P S,H and P S,H,max with respect to the diameter of the coil wire and with respect to the thickness of the conductive layer for a different electrical resistivity of the conductor material. Hence, by choosing the appropriate coil type, U S and I S can be influenced without changing the heating power due to a different inductance L S and a different linked magnetic flux Ψ PS .
Within the parameter sets considered in this work, an optimization with respect to maximize the heating power is carried out. Regarding this optimization, maximum heating power is achieved for d W = d L = 0.02 mm, ρ c = 325 nΩ · m (WWC), and ρ c = 375 nΩ · m (FWC) in case no reactive power compensation is used, whereas, according to (21), the lowest value of ρ c in the parameter set (ρ c = 16 nΩ · m) reveals maximum heating power for d W = 0.11 mm (WWC with litz wire) and d L = 0.20 mm (FWC with 10 windings) in case of a serial compensated secondary coil. With respect to the frequency of the alternating magnetic field and to the relative permeability of the core, the maximum values of these parameters in the parameter sets (f = 100 kHz and µ r = 2100) reveal the maximum heating power. The results of this evaluation are presented in Fig. 10. In Table 2, a summary on the parameters for This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. heating power of a serial compensated FWC increases to a maximum of P S,H ≈ 1.3 W at d L = 0.40 mm. For more than 80 windings, the heating power does not increase further. However, the single layers of foil have to be extremely thin for the realization of such a high number of windings. According to |Ψ PS | ∝ H P and P S,H ∝ |Ψ PS | 2 , the maximum heating power can be determined for any magnitude H P of the primary alternating magnetic field for all coil types with where P S,H,1 denotes the resulting heating power for H P = H 1 = 1 A m . P S,H,1 is presented in Table 2 for all coil types and for all types of reactive power compensation analyzed in this publication. Fig. 11 presents the dependencies of the maximum heating power on the electrical resistivity of the conductor material, on the frequency of the alternating primary magnetic field, and on the relative permeability of the core along with the corresponding diameter of wire and the corresponding thickness of layer for achieving maximum heating power for H P = 50 A m . In case of an uncompensated secondary coil, a particular electrical resistivity exits for achieving maximum heating power. For an increasing ρ c , the maximum heating power decreases and the diameter of wire (WWC) and the thickness of layer (FWC) increases. For decreasing ρ c , the maximum heating power rapidly decreases, whereas d W and d L remain constant at 0.02 mm due to this is the lowest value in the parameter set analyzed in this work. In case of a WWC, d W rapidly increases to 0.64 mm for a low electrical resistivity of the conductor material. For serial compensated secondary coils, the maximum heating power is achieved for the lowest electrical resistivity in the parameter set. d W and d L increase for increasing ρ c and converge to d W = 0.42 mm (WWC) and d L = 0.40 mm (FWC) for high ρ c . In case of increasing frequency, the maximum heating power for all coil types and compensation types increases as well, whereas the diameter of wire and the thickness of layer decrease for uncompensated secondary coils and remain constant at d W = 0.11 mm (WWC) and at d L = 0.20 mm (FWC) for secondary coils with serial reactive power compensation. The maximum heating power increases with increasing relative permeability of the core for uncompensated secondary coils and converges to P S,H ≈ 1.3 mW (WWC and FWC) for very high values of µ r . For serial compensated secondary coils, the heating power increases as well with increasing µ r and converges to P S,H ≈ 220 mW (WWC) and P S,H ≈ 240 mW (FWC) for very high values of µ r . The diameter of wire and the thickness of layer slightly decrease to and remain constant at 0.02 mm for µ r > 300 in case of an uncompensated secondary coil, whereas, for secondary coils with serial reactive power compensation, d W slightly increases and converges to 0.12 mm (WWC) and d L increases and converges to 0.22 mm (FWC) for very high values of µ r .

VII. PERFORMANCE
With respect to tumor treatment, the heat generated by an implanted secondary coil with a specific heating power and the resulting rise of temperature in the surrounding tissue has to be known. Additionally, unwanted tissue heating outside of the tumor by eddy currents has to be taken into account for ensuring the safety of the patients. In this section, a thermal evaluation is performed and the resulting SAR is evaluated. Furthermore, the PTE and the coupling coefficient with respect to two configurations of primary coils are determined and the impact of an angular or lateral misalignment of the secondary coil as well as the impact of body tissue on heating power and on coil design are discussed.

A. THERMAL EVALUATION
For determining the minimum heating power, which is required to heat human tissue above the necrosis temperature of 50 • C [14], [46] for being able to conduct a thermal tumor ablation, a thermal evaluation is carried out. Based on this minimum heating power, the minimum magnitude of the primary magnetic field strength is determined for all coil types and all compensation types with respect to the optimized parameter set, which reveals maximum heating power.
The thermal evaluation is done by a FEM simulation. A spherical area of human tissue with a diameter of 200 mm is used in the model. The cylindrical secondary coil is represented by a cylinder surrounded by a heat generating layer with a total diameter of d max and is placed in the center of the sphere. The thermal properties of copper [47] are assigned to the surrounding layer which are assumed to be similar to the thermal properties of a conductor material with a different electrical resistivity. Furthermore, the typical thermal properties of a ferrite material [48] are assigned to the core. As done in [49], the thermal properties of the tumor are defined to be similar to the surrounding tissue. A kidney tumor is assumed due to this represents a challenging scenario with respect to tissue heating based on the combination of a high specific heat capacity, a high thermal conductivity and a high density. All thermal tissue properties are taken from [50]. The thermal conductivity and the heat capacity of tissue depend on water content [51]- [53]. Hence, based on the desiccation of tissue with rising tissue temperature [54], the thermal conductivity and the heat capacity decrease with increasing temperature. Therefore, these thermal properties are reduced linearly between 50 • C and 100 • C. The resulting thermal material properties are summarized in Table 3. On  the outside boundaries of the spherical area of human tissue, a constant temperature of 37 • C is defined to represent the thermal interface to the surrounding part of the human body, which keeps the boundaries at body temperature. However, as the surrounding part of the body is heated as well in real-life thermal tumor ablations, this is a worst case scenario with respect to increasing tissue temperature for simplifying the thermal FEM model. As shown in Fig. 12, a heating power of P S,H = 1.5 W reveals a tissue temperature above the necrosis temperature in a distance up to 7 mm in the axial direction and in a distance up to 13 mm in radial direction to the cylindrical secondary coil. Thus, a spheroidal tumor with 34 mm in length and 27.5 mm in width can be ablated. Furthermore, a maximum tissue temperature of 114 • C is achieved by a heating power of 1.5 W. The resulting required magnitude of the primary magnetic field strength for all optimized coil types and compensation types is presented in Table 4 along with a summary of the results of the thermal evaluation.

B. SPECIFIC ABSORPTION RATE
With respect to unwanted heating of healthy tissue, the SAR is the key parameter for the evaluation of the influence of eddy currents on tissue temperature. The SAR is defined as where J denotes the magnitude of the local current density within the body tissue, ρ dens,B denotes the density, and σ B denotes the electrical conductivity of the body tissue. In [42], different primary coil configurations are analyzed and compared and the resulting SAR for each coil configuration is evaluated at a frequency of f = 100 kHz based on a model of the human body with respect to unwanted tissue heating. In case the frequency of the primary alternating magnetic field, the electrical conductivity, and the density of the body tissue remain constant, the resulting SAR can be calculated for each primary coil system with based on the magnetic field strength H P needed for generating the required heating power evaluated in Section VII-A (P S,H = 1.5 W), which is necessary for achieving an adequate rise of tissue temperature. In (26), H ref and SAR ref can be taken from [42] (Table VIII, column H TA,av and column SAR). The resulting SAR for a single longitudinal primary coil (SiCLoC) and an optimized Helmholtz primary coil (DCHC) is presented in Table 4. For these coil types, the SAR remains below the limit defined in [55] (2 W kg ) for uncompensated as well as for serial compensated secondary coils. This limit is considered to not cause a rise of tissue temperature and therefore to not cause physiological stress for patients.

C. POWER TRANSFER EFFICIENCY AND COUPLING COEFFICIENT
As done in Section VII-B, the PTE and the coupling coefficient between the primary coils and the implanted secondary coil is evaluated based on the primary coil configurations analyzed in [42]. For determining the PTE, 10 windings are assumed for each single coil of the primary coil configurations. The results are presented in Fig. 13. The coupling coefficient shows a negligible dependency on the type of secondary coil and on the number of windings of FWC. The maximum PTE is achieved for the same diameter of wire d W and for the same thickness of layer d L on which the heating power is at maximum, due to a low coupling between the primary coils and the secondary coil and therefore a primary current with constant magnitude is assumed. The low coupling is confirmed by the evaluation of the coupling coefficient k in Fig. 13c, which reveals a maximum of 14 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and  Table 4. For secondary coil types with serial reactive power compensation, the maximum PTE and the maximum coupling coefficient are not achieved at the same diameter of wire or the same thickness of layer, respectively.

D. MISALIGNMENT
In the analysis done in this publication, the field lines of the homogeneous primary magnetic field are assumed to be parallel to the axis of the core of the secondary coil. In case of an angular misalignment of the secondary coil with respect to the field lines of the primary magnetic field, the magnitude H || of the primary magnetic field component, which is parallel to the axis of the coil, can be approximated with where α denotes the angle between the field lines and the axis of the coil core. As a coil core with µ r > 1 influences the spatial distribution of the primary magnetic field, a field component, which is parallel to the core of the coil, exists even for α = 90 • . This parallel field component can be neglected for the dimensions of the secondary coil assumed in this paper. The impact of a lateral misalignment depends on the spatial distribution of the primary magnetic field and hence depends on the specific primary coil configuration. For all coil configurations analyzed in [42], a nearly homogeneous magnetic field is generated within a spherical area with a diameter of 40 mm surrounding the implanted coil. Therefore, the impact of a lateral misalignment on the heating performance can be neglected within this area for all primary coil configurations presented in [42].

E. IMPACT OF BODY TISSUE
In contrast to the thermal evaluation in Section VII-A, the impact of the tissue surrounding the implanted coils is not taken into account by the numerical-analytic calculation of the heating power as the influence of the body tissue on the heating power, the coil properties, and the coil design has shown to be negligible. This is due to the maximum electrical conductivity and the relative permeability of human tissue are negligibly low compared to the electrical conductivity of the conductor material and the relative permeability of the coil core. Additionally, within the frequency range considered in the present paper, the impact of the patient's body tissue on the alternating primary magnetic field caused by eddy currents induced in the body tissue turned out to be negligibly low as well.

VIII. DISCUSSION
The results of the numerical-analytic analysis done in this paper shows the possibility to significantly enhance the performance of a contactless thermal tumor ablation by choosing an appropriate design of the heat generating implanted coil. In most cases, FWC yields more heating power compared to WWC based on the same diameter of wire or thickness of layer, respectively. However, by choosing a conductor material with an appropriate electrical resistivity, the same heating power can be achieved with uncompensated WWC and FWC, as shown by P S,H,max in Fig. 10, whereas a serial compensated FWC provides more heating power than a WWC with serial reactive power compensation. Regarding the mechanical construction of the implanted coils, WWC requires more construction effort compared to FWC due to the number of windings is much higher. Additionally, the short circuit connection needs to be longer and causes a slight deformation of the single windings, which has not been taken into account in this analysis.
The heating power of different types of ferromagnetic implants have been analyzed and optimized in the last decades. A maximum heating power per implant length in the range from approximately 18 W m to 40 W m have been reported by the authors in [39], [40] for heat generation by eddy currents within the implant and approximately 47 W m have been achieved by adding a metallic sheath surrounding a ferrite core in [41]. An uncompensated optimized implanted secondary coil proposed in this paper shows less maximum heating power per implant length (approximately 37 W m ) compared the results shown in [39] (approximately 40 W m ) for equal conditions (H P = 1.5 kA m peak, f = 100 kHz, µ r = 150). Moreover, the implant diameter in [39] is 0.1 mm less (1.4 mm) than the implant diameter used in the present publication. In contrast to this, an uncompensated optimized implanted secondary coil (FWC) reveals about 2.4 times more heating power per unit length (113 W m ) compared to the results reported in [41] (47 W m ) for a ferrite core surrounded by a metallic sheath with an overall diameter of 1.5 mm and a length of 24.85 mm (H P ≈ 1.5 kA m rms, f ≈ 100 kHz, µ r ≈ 2000 at 50 • C).

IX. CONCLUSION
With the numerical-analytic analysis carried out in the present publication, a way for precisely and efficiently evaluating the heating power of implanted wire wound and foil wound coils with restricted dimensions for a thermal tumor ablation of deep-seated tumors is presented. Based on the results of this paper, the optimum design of the implanted coil can be determined with respect to different limiting conditions, such as a specific diameter of the conductor or a specific operating frequency. However, the thermal model is kept simple and the influence of blood flow is not taken into account in the thermal analysis. Moreover, the resistance of the short circuit connection of the coils and the ESR of the compensating capacitor are solely considered to some extent with constant values and hot spot generation on these resistances is not taken into account in the thermal analysis. Additionally, the spatial requirements and the inductance of the short circuit connection and the compensating capacitors are assumed to be negligible small. Subsequent to this paper, a precise experimental measurement setup has to be realized for an additional validation of the results achieved in this work.
The SAR is kept well below the limits published in international standards with uncompensated and serial compensated implanted coils. Hence, an appropriate rise of temperature within the tumor can be achieved while preventing to cause physiological stress for patients by heating healthy tissue. Generally, by maximizing the ratio of heating power to required primary magnetic field strength, the risk of unwanted heating and influencing of other implanted devices, such as pacemakers, artificial joints, screws, plates, clips, and stents, is minimized, which enhances the safety and the well-being of the patients as well as the outcome of the tumor treatment.