Eddy Current Heating of Implanted Devices for Tumor Ablation: Numerical-Analytic Analysis and Optimization

Millimeter-sized devices, which are implanted within tumors by a minimally invasive operation technique, are very promising for tumor treatment with a contactless thermal ablation procedure. These implanted devices are heated from outside of the patients’ body by an alternating magnetic field. For minimizing unwanted influencing and heating of healthy tissue and other devices inside the patients’ body, the ratio of generated heating power within the implanted devices to required magnetic field strength has to be maximized. In this paper, the heating power generated by eddy currents within solid and electrically conductive implanted devices, whose dimensions are restricted by the minimally invasive operation technique, is analyzed and optimized based on a combination of numerical and analytic calculations for different parameter sets of material and magnetic field properties. The parameters for achieving maximum heating power are presented along with the dependency of the heating power on these parameters. The results are validated by experimental prototype measurements. Furthermore, the specific absorption rate is evaluated based on specific coil configurations for generating the alternating magnetic field. This paper shows the feasibility of significantly increasing the heating power of solid and electrically conductive implanted devices by choosing the appropriate properties of the implant material. This enhances the safety and the well-being of the patients and represents a great benefit for the outcome of the tumor treatment.


I. INTRODUCTION
Different procedures for the treatment of tumors are established in clinical practice. Compared to invasive open tumor surgery, non invasive and minimally invasive tumor treatment procedures are very advantageous since the stress for patients is reduced significantly. Depending on the type, the localization and the extent of a tumor, a tumor removal may not be possible. For these cases, a tumor ablation is advantageous. For metastatic solid tumors, mostly a palliative situation is given, which solely focuses on reducing the tumor growth rate and on reducing the tumor spreading instead of healing the tumor. One option for conducting a contactless The associate editor coordinating the review of this manuscript and approving it for publication was Valentine Novosad. thermal tumor ablation procedure is represented by a single implantation of inductively heated millimeter-sized objects into the tumor by a minimally invasive operation technique. The implanted devices are intended to remain within the tumor, which enables an unlimited repetition of the tumor treatment and which extends the maximum possible time of application without the need of additional surgical interventions and without additional stress for the patients.
In the field of non invasive and minimally invasive thermal tumor ablation (hyperthermia), contactless energy transfer (CET) is used for heating up and ablating tumor tissue. Heat is generated directly within tumor tissue or within devices, which are injected or implanted directly into the tumor and heat up the corresponding tissue, by ohmic or magnetic losses based on alternating magnetic fields [1], [2], [3], [4]. Heating up tumor tissue indirectly by generating heat in permanently implanted devices is also referred to as magnetically mediated hyperthermia (MMH) [5], [6].
The contactless heating of tumor tissue by implanted devices based on an inductive transfer of heating energy has been discussed and researched intensively in the last decades. A general analysis of hyperthermia with inductively heated implanted devices was carried out in [7] and the temperature distribution within the tissue, which is heated by implanted devices, was analyzed by the authors in [8], [9], and [10]. In [11], the magnetic power absorption caused by hysteresis losses was calculated and a new mathematical formula for the approximation of magnetic heating power was proposed. An entire system for inductive hyperthermia comprising four primary coils placed on a ferromagnetic core has been proposed and analyzed in [12] and the heating performance of an implanted device consisting of a two-sectional needle comprising a stainless steel and a ferrite material was evaluated. Numerical models for calculating the heating power or the eddy current density of inductively heated implanted devices were presented in [13] and [14]. The heating power of ferromagnetic implanted devices was evaluated for specific combinations of parameters by the authors in [15], [16], and [17], such as magnitude and frequency of the magnetic field, based on analytic equations. Here, an implanted device with infinite length was assumed. In [18], the end effects of a cylindrical implanted device with finite length were analyzed and a ferrite core surrounded by an electrically conductive layer was analyzed by the authors in [19]. In [20], the heating performance of coil type implanted devices was evaluated for uncompensated coils and for coils with reactive power compensation. Moreover, different inductive links and coil type implants were investigated with respect to a four-coil system by the authors in [21] and [22] considering reactive power compensation with respect to the Q-factor. Additionally, a performance optimization has been done. The maximum deliverable heating power under a specific absorption rate (SAR) constraint and under a specific power transfer efficiency (PTE) constraint was investigated in [23] with respect to coil type implanted devices with reactive power compensation. The power absorption and the heating performance were analyzed and optimized for implanted coils with respect to different parameters, objectives and constraints by the authors in [24], [25], [26], and [27], whereas the heating power per unit volume of infinitely long ferromagnetic implanted devices was analyzed in [28], [29], and [30]. Additionally, an optimization of the heating power per unit volume has been carried out based on the optimum induction number. For this, a solid implanted device was replaced by multiple strands of wire fitting to the same cross sectional area.
In the present publication, the heating power of a solid ferromagnetic and electrically conductive implanted device, which is generated by an alternating magnetic field and corresponds to the power delivered to load (PDL), is evaluated and analyzed based on numerical calculations with the finite element method (FEM) and based on analytic calculations. The implanted device is assumed to be cylindrical and the dimensions are chosen to fit to the spatial requirements of a minimally invasive operation technique for positioning the implanted device in the tumor. For the evaluation of the heating power, the influence of the implanted device on the spatial distribution of the alternating magnetic field is determined precisely by taking into account the dimensions and the magnetic properties of the material of the implanted device. Furthermore, the influence of the end effects originating from the finite length of the implanted device on the heating power is precisely taken into account as well. An optimization for achieving maximum heating power is done with respect to the relative permeability and electrical resistivity of the material of the implanted device as well as with respect to the frequency of the alternating magnetic field. The dependency of all single parameters on each other for achieving maximum heating power is analyzed and presented along with the theoretically achievable maximum heating power. Moreover, the SAR is determined based on two coil configurations for generating the alternating magnetic field. A comprehensive and precise analysis of the influence of material and magnetic field properties on the heating power of a solid ferromagnetic and electrically conductive implanted device with finite length and a comprehensive maximization of the heating power have not been done in scientific literature yet.
In Section II and Section III of this paper, the settings of the numerical-analytic analysis and the modeling of eddy currents are presented. The structure of the numerical-analytic analysis is introduced and the numerical and the analytic calculations are described in detail in Section IV followed by the presentation of the results of the analysis and by an evaluation of the heating performance, of the SAR as well as of the impact of body tissue. Subsequently, the analysis results are validated by experimental prototype measurements in Section VII and discussed in Section VIII. Finally, some concluding words are given in Section IX.

II. ANALYSIS SETTINGS
The heating power created in a solid and electrically conductive implanted device by an alternating magnetic field depends on different parameters, such as the dimensions and the material properties of the implanted device. Furthermore, the direction and the spatial distribution of the alternating magnetic field with respect to the implanted device have a significant impact on the heating power.
For the analysis carried out in the present publication, a contactless thermal tumor ablation procedure based on heating implanted devices by eddy currents is assumed. This implanted device is positioned within the tumor by a minimally invasive operation technique. Hence, the dimensions of the implanted device are restricted to fit to a narrow tube (trocar), through which the implanted device is placed within the tumor. Accordingly, the shape of an implanted device is assumed to be cylindrical with a diameter of d I = 1.5 mm and a length of l I = 20 mm. As the worst case scenario of deep-seated tumors is assumed in this analysis, the heating energy has to be transferred over a distance of 25 cm in case the diameter of a human body is 50 cm. Therefore, the diameter of the coils for generating the alternating magnetic field has to be considerably large compared to the dimensions of the implanted device. Based on this, the alternating magnetic field is assumed to be homogeneous and the magnetic field lines are assumed to be parallel to the axis of the implanted device in the area surrounding the implanted device. Due to the considerably large distance between primary field generating coils and the implanted device and due to the large difference in dimensions of the implanted device compared to the primary coils, an extremely low coupling is expected. Hence, the magnitude of the current applied to the primary coils is assumed to be constant. This enables to analyze the heating power created in the implanted device based on the properties of the alternating magnetic field without considering the type and the properties of the primary coils.
The magnetic and electrical properties of the material of the implanted device as well as the magnitude and the frequency of the primary alternating magnetic field significantly influence the heating power. For evaluating the optimal properties of the implanted device and the primary magnetic field to achieve maximum heating power and for analyzing the dependency of the heating power on these properties, parameter sets for the relative permeability µ r and the electrical resistivity ρ of the material of the implanted device as well as for the frequency f of the primary alternating magnetic field are defined. These parameter sets are presented in Table 1 along with the corresponding skin depth δ. The relative permeability is assumed to be independent of the magnitude and of the frequency of the primary magnetic field. As the skin effect strongly depends on the magnetization curve, for nonlinear materials, an evaluation of the heating performance and of the optimal parameters has to be done separately for each individual material comprising an individual magnetization curve.

III. MODELING OF EDDY CURRENTS
A current is induced in a cylindrical implanted device in ϕ-direction of a cylindrical coordinate system in case the z-axis corresponds to the axis of the implanted device by an alternating magnetic field, which is parallel to the axis of the implanted device. Hence, a solid and electrically conductive implanted device represents the secondary side of an inductive CET system. Accordingly, as done in [27] for wire wound coils (WWC) and for foil wound coils (FWC), a solid and electrically conductive implanted device can generally be represented by a voltage source, an inductance L S , and a resistance R S as shown in Fig. 1 for calculating losses created by an alternating magnetic field. Thus, the current within the implanted device can be expressed as where Ψ PS denotes the linked magnetic flux, which is received from the primary coils, and ω denotes the angular frequency of the alternating magnetic field. In the analysis done in the present paper, a current with a constant magnitude is assumed to be applied to the field generating coils. This results in a primary magnetic field with a constant magnitude and therefore in a constant magnitude of Ψ PS . The number of turns for the model of this type of implanted device results in N S = 1. The penetration depth (skin depth) δ depends on the relative permeability µ r and the electrical resistivity ρ of the implant material as well as on the frequency f of the primary magnetic field. For the magnitude of the induced current density is reduced to the e-th part of the current density magnitude on the surface of the implanted device [31]. Hence, the induced current is limited to a specific region within the implanted device. A specific inductance L S and a specific resistance R S , which heats up the implanted device, can be assigned to this specific current density created by an alternating magnetic field, which is represented by a voltage source as shown in Fig. 1.

IV. NUMERICAL-ANALYTIC ANALYSIS
For evaluating the heating power of an implanted device created by eddy currents for the parameter sets defined in Section II, a numerical analysis is carried out. Subsequently, based on the numerical results, an analytic analysis is done. This reduces computational costs and increases the speed and the flexibility of the entire numerical-analytic analysis due to the numerical results can be reused for the analytic calculations, which are less time consuming.

A. NUMERICAL CALCULATIONS
The influence of the ferromagnetic implanted device on the spatial distribution of the primary magnetic field, which depends on the relative permeability µ r of the implanted device, and on the current density #» J S within the implanted device, which depends on the spatial distribution of the magnetic field and on the skin depth δ, can be expressed analytically in terms of elliptic integrals or Bessel functions. These have to be approximated by series expansion or evaluated by numerical calculations. As no exact analytic expression in terms of elementary functions is feasible, the influence of the ferromagnetic implanted device on the spatial distribution of the magnetic field and the corresponding current density within the implanted device are evaluated with FEM in the present publication.
Based on the rotational symmetry of the cylindrical implanted devices, solely 2D calculations are conducted in the rz-plane of a cylindrical coordinate system at ϕ = 0 to reduce computational costs. For the evaluation of the results, a rotation around the middle axis of the implanted device is done. A homogeneous alternating magnetic field with a magnitude H 0 = 1 A m , which is parallel to the axis of the implanted device, is applied by assigning appropriate boundary conditions to the FEM model.
Generally, for a constant implant diameter and a constant implant length, L S and R S depend on the magnetic and electrical properties of the implanted device as well as on the frequency of the primary magnetic field, whereas Ψ PS solely depends on the magnetic properties of the implanted device. With respect to the skin depth δ, L S and R S show a dependency on δ for all appropriate frequencies and electrical resistivities as well as on the relative permeability µ r of the implant material. L S is defined by a current within the implanted device and by the resulting magnetic flux Ψ S , which is created by this current and depends on δ. Hence, the linked magnetic flux Ψ PS , which is created by the primary side and received by the implanted device, does not necessarily be dependent on δ as well. According to the results of the analysis done in the present work, Ψ PS shows to be independent of δ and solely depends on µ r . Hence, the numerical calculations are done for the parameter sets of δ and µ r . This enables to reuse the results in the analytic calculations for any combination of frequency and electrical resistivity revealing the same skin depth.
The heating power P S,H,0 and the current within the implanted device I S,0 , which is created by the alternating magnetic field with the magnitude H 0 , are evaluated for a fixed electrical resistivity ρ 0 = 16 n · m, which corresponds to the electrical resistivity of silver and represents the lowest value in the parameter set of ρ. By choosing the appropriate frequency f 0 of the alternating magnetic field, the requested skin depth is achieved. Additionally, the corresponding linked magnetic flux Ψ PS,0 received from the primary side is calculated numerically. For this, the component of the magnetic flux density, which is parallel to the axis of the implanted device, is averaged within the implanted device and multiplied by the cross sectional area, which is normal to the axis of the implanted device. This precisely reveals Ψ PS,0 .

B. ANALYTIC CALCULATIONS
Based on the results of the numerical calculations, the heating power P S,H generated in the implanted device by an alternating magnetic field created from the primary side with the magnetic field strength H P is determined analytically for the parameter sets of the relative permeability and the electrical resistivity of the implanted device and for the parameter set of the frequency of the primary alternating magnetic field. For this, the resistance R S,0 , which corresponds to ρ 0 , is determined analytically with Subsequently, L S and R S are evaluated with and Here, ω 0 denotes the angular frequency corresponding to f 0 . VOLUME 11, 2023 Due to Ψ PS ∝ H P , Ψ PS results in Hence, the heating power generated in the implanted device can be determined analytically for any primary magnetic field strength with According to (7), the theoretically achievable maximum heating power results in for a conductor resistance For evaluating the heating power P S,H,x for a specific relative permeability µ r,x and a specific electrical resistivity ρ x of the implanted device as well as for a specific frequency f x and a specific magnetic field strength H P,x of the primary magnetic field, the corresponding skin depth δ x is determined. Subsequently, the resulting R S,x and L S,x are evaluated based on δ x and based on the results of the numerical calculations. Finally, P S,H,x is calculated with (6) and (7). The structure of the analytic calculations is shown in Fig. 3.

V. RESULTS
For maximizing the heat generated in the implanted device, the heating power is calculated for all parameter sets as presented in Section IV and the maximum heating power is determined along with the corresponding values in the parameter sets. The dependency of P S,H on the relative permeability and the electrical resistivity of the implant material and on the frequency of the alternating primary magnetic field is evaluated along with the dependency of the parameters revealing maximum heating power on each other. Additionally, the theoretical achievable maximum heating power is evaluated.

A. ACHIEVABLE MAXIMUM HEATING POWER
In Fig. 4, R S and L S resulting from numerical calculations are presented depending on the relative permeability and on the skin depth along with |Ψ PS |, which solely depends on µ r . For being able to choose the skin depth and the relative permeability independently, for every combination of skin depth and relative permeability shown in Fig. 4a and in Fig. 4b, the frequency is adjusted for revealing the corresponding specific skin depth and the corresponding specific relative permeability. R S decreases with increasing skin depth and slightly increases with increasing relative permeability, whereas L S shows an increase for an increasing skin depth and for an increasing relative permeability, especially for high values of δ. The linked magnetic flux increases rapidly for low values of µ r up to µ r ≈ 500 and converges to |Ψ PS | ≈ 165 pWb for very high values of µ r .
According to the results of the numerical and analytic calculations, a maximum heating power of P S,H = 537 µW, which corresponds to the PDL, is achieved for ρ = 347 n ·m and the maximum values in the corresponding parameter sets for the frequency of the primary magnetic field and for the relative permeability of the implant material (f = 100 kHz, µ r = 2100) in case of an alternating primary magnetic field with H P = 50 A m is applied to the implanted device. This is summarized in Table 2 along with the corresponding values for R S and L S .
In Fig. 5, the results of the heating power evaluation are presented and the dependency of the heating power on the electrical resistivity and the relative permeability of the implanted device and on the frequency of the primary alternating magnetic field is shown. In case of an increasing frequency, the heating power increases as well for all remaining parameters in the parameter sets considered in this publication. The heating power maximizes for a specific electrical resistivity ρ or for a specific relative permeability µ r , respectively, in case the remaining parameters are considered to be constant. The values of ρ and of µ r corresponding to the maximum values of the heating power are evaluated depending on µ r or depending on ρ, respectively, as well as depending on the frequency of the primary alternating magnetic field along with the corresponding maximum heating power. The results of this evaluation are presented in Fig. 6. For a low relative permeability of µ r ⪅ 500 and for a frequency in the upper range of the parameter set considered in this paper, a high electrical resistivity has to be chosen for the implant material to maximize heating power. Outside this area, the heating power is maximized by using electrically high conductive materials for the implanted devices. Accordingly, the maximum heating power for electrically low conductive implanted devices is achieved by a low relative permeability. For example, ρ = 3 µ · m at f = 100 kHz and at H P = 50 A m leads to a maximum heating power of P S,H = 0.499 mW, which is about 93% of the overall maximum achievable heating power, for a considerably low relative permeability of µ r = 373.
In [28], [29], and [30], the heating power of ferromagnetic implanted devices are analyzed based on the induction number where d I denotes the diameter of the implanted device. The induction number for an implanted device with optimized VOLUME 11, 2023 52093 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. parameters with respect to maximum heating power yields i = 51.8 based on the results of the present publication.

B. THEORETICAL MAXIMUM OF THE HEATING POWER
As shown in Section IV-B, the maximum heating power P S,H,max , which can theoretically be achieved in case the resistance R S = R S,max (see (9)) can be chosen independently without influencing the skin depth by adjusting ρ and therefore without influencing L S , can be determined with (8). As summarized in Table 2, the maximum theoretically achievable heating power maximizes at P S,H,max = 1.58 mW (H P = 50 A m ) for the parameter sets analyzed in this paper in case R S is fixed at 3.57 m and the skin depth is fixed at the lowest value δ min = 4.39 µm in the parameter set. The corresponding relative permeability of the implant material and the corresponding frequency of the primary magnetic field result in µ r = 806 and f = 100 kHz, hence the related inductance of the implanted device results in L S = 5.69 nH. Since R S and δ are fixed, the electrical resistivity of the implant material does not influence the results. This cannot be realized by a solid and electrically conductive implanted device due to R S and L S depend on each other by influencing the skin depth. A practical way for adjusting the implanted device to increase the heating power beyond the maximum heating power achieved by a solid and electrically conductive implanted device is proposed and discussed in Section VIII of this publication.
The theoretically achievable maximum heating power is presented in Fig. 7 for different relative permeabilities and for different frequencies and the parameters for the overall theoretical maximum heating power are summarized in Table 2.
For an increasing frequency of the primary alternating magnetic field, P S,H,max increases as well, whereas a specific frequency dependent relative permeability exists, for which P S,H,max maximizes.

VI. PERFORMANCE EVALUATION
For analyzing the performance of generating heating power based on eddy currents in a solid and electrically conductive implanted device, the ability to rise tissue temperature and the 52094 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  corresponding required heating power as well as the required magnetic field strength of the primary magnetic field have to be considered. Furthermore, the SAR has to be taken into account. The results of the performance evaluation are presented in Table 3 for an implanted device optimized to generate maximum heating power for a given specific magnetic field strength.

A. TISSUE HEATING
According to the thermal analysis conducted in [27] based on a basic model of the human body and based on thermal parameters, which represent the worst case scenario with respect to the heating of tumor tissue, a heating power of 1.5 W is necessary to achieve an adequate rise of tissue temperature for conducting a thermal tumor ablation. A spheroidal ablation area of 34 mm in length and 27.5 mm in width with a tissue temperature of or above the necrosis temperature of 50 • C [2], [32] up to a maximum temperature of 114 • C is realized by an implanted device with the dimensions analyzed in the present paper. Based on the results of the numerical-analytic analysis of the heating power presented in Section V-A, a magnetic field strength of H P = 2642 A m is required for achieving a heating power of P S,H = 1.5 W by an optimized implanted device.

B. SPECIFIC ABSORPTION RATE
The unwanted heating of healthy body tissue by inducing eddy currents is taken into account by the SAR. Hence, the SAR represents a key parameter for evaluating the safety of a contactless thermal tumor ablation based on an inductive transfer of the heating energy. The SAR is evaluated based on two primary coil configurations for generating the alternating magnetic field based on the results of the analysis done by the authors in [33], in which different coil configurations are analyzed and compared with respect to conduct a contactless thermal tumor ablation. With where H ref and SAR ref can be taken from [33] (Table VIII, column H TA,av and column SAR), the SAR for a solid and electrically conductive implanted device optimized for achieving maximum heating power is calculated for a single longitudinal primary coil (SiCLoC) and eight sagittal primary coils (OCC). The SAR results in 0.715 W kg (SiCLoC) and 1.41 W kg (OCC), which is below the limit defined in [34] (2 W kg ). Thus, the safety of the patients is essentially increased by the maximization of the heating power for a specific magnetic field strength as this limit is considered to not cause a rise of tissue temperature and therefore to not cause physiological stress for patients.

C. IMPACT OF BODY TISSUE
The impact of body tissue on the heating power shows to be negligible, as the electrical resistivity of tissue is considerably higher than the electrical resistivity of the implanted device considered in this analysis and the relative permeability of tissue is assumed to be µ r = 1, which approximates the relative permeability of water. Additionally, the magnetic fields created by eddy currents, which are induced directly in the body tissue, turned out to be negligibly low as well and hence do not affect the spatial distribution of the primary alternating magnetic field in the frequency range considered in this analysis.

VII. MEASUREMENT VALIDATION
For validating the results achieved by the numerical-analytic analysis done in the present paper, multiple implanted device VOLUME 11, 2023 FIGURE 8. Electrical equivalent circuit of the experimental prototype measurement setup, which represents the primary side of an inductive CET system with reactive power compensation.
prototypes comprising different specific sets of material properties are manufactured. These prototypes are exposed parallel to the field lines of an homogeneous alternating magnetic field with a constant magnitude H P,m and the power loss within the prototype is measured.

A. HEATING POWER EVALUATION
The primary side of an inductive CET system with serial reactive power compensation can be represented by an inductance L P , by a resistance R P , by a capacitor C P , and by an impedance Z ′ S . Energy is supplied to the primary side by a voltage or current source with the voltage U Gen and the current I P , respectively. Fig. 8 shows the electrical equivalent circuit for the experimental prototype measurement setup, which represents the primary side of an reactive power compensated inductive CET system. Z ′ S describes the influence of the secondary side, which is the implanted device in the present analysis, on the primary side including the heating power and coupling. The power loss created by the resistance R S of the implanted device (see Fig. 1) corresponds to the heating power and is equal to the power loss created by the equivalent resistance R ′ S on the primary side, which can be determined with Hence, in case a current I P with constant magnitude is applied to the primary side, the power loss created on the primary side by the resistance R P is constant as well. Accordingly, the heating power P S,H can be determined by evaluating the increase of input power applied to the primary side when exposing an implanted device to the alternating magnetic field with P S,H = P P − P P,0 , where P P,0 denotes the primary side input power without implanted device and P P denotes the primary side input power in case an implanted device is exposed to the alternating magnetic field.

B. MEASUREMENT SETUP
In the analysis done in this work, a current with a constant magnitude is assumed to be applied to the primary coils. This enables to evaluate and to maximize the heating power generated in the implanted device based on the magnetic field strength applied to the implanted device independent of the type and the properties of the primary coils. Hence, for the experimental prototype measurement setup, small scale primary coils in a nearly ideal Helmholtz configuration are used for generating a homogeneous magnetic field with an adequate magnetic field strength. Each single coil comprises 30 turns consisting of an appropriate litz wire with a diameter of 70 mm and a wire diameter of 0.9 mm. A magnetic field strength of H P,m = 653 A m is measured on the axis of the coils in the middle between both coils in case a current of I P,m = 1 A is applied to the coils. The setup of the experimental prototype measurement is shown in Fig. 9.
As shown in Fig. 8 and Fig. 9, a series resonant circuit consisting of C P and L P is used for generating a sinusoidal voltage U P and the corresponding sinusoidal current I P within the primary inductance L P as well as for reactive power compensation. A rectangular voltage U Gen is applied to the series resonant circuit and the switching of U Gen is managed by a control unit. For the experimental prototype measurement, a sinusoidal current with a constant magnitude of 3 A and a frequency of 101 kHz is applied to the coils.

C. RESULTS
For the experimental prototype measurement, samples consisting of different types of stainless steel (A, B, and C) and consisting of soft iron (D) comprising different material properties and comprising dimensions corresponding to the implanted devices analyzed in this work are manufactured. The magnetization curve and hence the permeability of stainless steel and soft iron strongly depend on material composition and on material treatment, such as mechanical processing and annealing.
The primary side input power without implanted device results in P P,0 = 3.68 W for a primary current of 3 A. Hence, the heating power within the implanted device increases the primary side input power by approximately 16% to 25%. The waveforms and the corresponding derived values of a measurement are presented in Fig. 10. The results of the measurement are presented in Table 4 and show a reasonable match to the corresponding results of the numerical-analytic evaluation, which are as well presented in Table 4. However, especially for stainless steel, mainly approximate values or value ranges are provided in the corresponding data sheets with respect to the magnetic properties of the material.

VIII. DISCUSSION
The numerical-analytic analysis carried out in the present publication presents the feasibility to maximize the heating power generated based on eddy currents by cylindrical implanted devices with finite length by choosing appropriate material properties. Magnetic losses show to be low compared to eddy current losses and are therefore not considered in this analysis.
The theoretical maximum of the heating power P S,H,max , which is presented and evaluated in Section V-B, cannot be achieved by solid and electrically conductive implanted devices. The inductance L S and the resistance R S in the electrical equivalent circuit of the implanted device depend on the skin depth. For achieving P S,H,max , R S = ω · L S has to apply. By adjusting the electrical resistivity of the implant material, the skin depth is changed and hence L S changes as well. A practical way to specify R S and L S independently is an electrically non-conductive cylindrical core, which is covered by an electrically conductive material comprising a relative permeability identical to the core material. Generally, foil wound coils, as analyzed in [27], are an opportunity to accomplish this. The achievable maximum heating power revealing from the present paper is 0.43 times and the theoretical maximum heating power revealing from the present paper is 1.25 times the maximum heating power presented in [27] for foil wound coils of the same dimensions as analyzed in the present work assuming a conductor material with a relative permeability of µ r = 1. Hence, implanted foil wound coils are generally capable of generating a higher heating power than solid and electrically conductive implanted devices with respect to identical implant dimensions and to an identical primary magnetic field. As the heating performance strongly depends on the spatial properties of the object intended to be heated, such as ratio of diameter to length in case of a cylindrical object, and on the orientation of the object with respect to the direction of the magnetic field lines, the requirements for an optimal heating performance of the specific implanted devices assumed in the present analysis may not be directly comparable to the requirements necessary for an optimal inductive heating performance in other fields of applications.
According to the results of the numerical-analytic analysis carried out in this work, P S,H,max maximizes for the minimum skin depth δ min = 4.39 µm resulting from the parameter sets assumed in this paper. However, for a further decreasing skin depth, the maximum of P S,H,max increases. With respect to foil wound coils, this can be done by reducing the thickness of the foil. This has as well been observed in [27] for a decreasing thickness of the conductor foil.
A maximum heating power per unit length of approximately 18 W m has been evaluated in scientific literature for implanted devices with a diameter of 1 mm and a length of 50 mm at H P = 1.62 kA m and f = 104 kHz in [30] as well as of approximately 12 W m at H P = 889 A m and f = 82.7 kHz for implanted devices with a diameter of 1.4 mm and an unspecified length by the authors in [29] based on measurements for stranded implanted devices. Furthermore, a heating power of 24 W m to 40 W m has been reported in [29] based on calculations for solid and stranded implanted devices with a diameter of 1.4 mm and an infinite length at H P = 1.5 kA m and f = 100 kHz. The analysis done in this work reveals a maximum heating power per unit length of 7 W m and 24 W m , respectively, for solid implanted devices as well as FIGURE 10. Waveforms and the corresponding derived values of a measurement of the heating power, which is generated within an implanted device. C1: Rectangular input voltage U Gen ; C4: Sinusoidal current I P within the primary coil; F1: Input power; P1: Magnitude of I P ; P2: Average input power P P,0 or P P , respectively. (a) Measurement without implanted device, (b) measurement with implanted device exposed to the magnetic field.
a theoretical maximum heating power per unit length of 21 W m and 71 W m , respectively, for implanted devices whose parameters in the electrical equivalent circuit can be specified separately evaluated at the corresponding applying conditions for the measurement and for the calculation in [29]. Hence, the numerical-analytic analysis done in the present paper shows a higher theoretically achievable maximum heating power than reported in [29] for optimized stranded implanted devices. However, the diameter of the implanted devices analyzed in [29] is 0.1 mm less, the length of the measurement prototypes is not specified, and an infinite length has been assumed for the calculations.

IX. CONCLUSION
In a contactless thermal tumor ablation procedure, it is essential to generate heat solely in the area of the tumors and to minimize unwanted heating and influencing of other implanted devices, such as pacemakers, artificial joints, screws, plates, clips, and stents by the primary alternating magnetic field. For maximizing the ratio of generated heating power to required magnetic field strength for solid and electrically conductive implanted devices, the heating power is analyzed in the present publication based on a numerical-analytic analysis for different parameter sets of material and magnetic field properties and the parameters resulting in maximum heating power for a specific field strength of the alternating magnetic field are presented. However, the relative permeability of the implant material is assumed to be independent of the magnetic field strength and no saturation effects within the implanted device are taken into consideration. Additionally, the feasibility of implementation of the optimized material properties is not investigated in this publication. This has to be done in future works with focus on material design.
The present work shows the feasibility to maximize the heating power of implanted devices with respect to specific constraints of material or magnetic field properties. The heating power maximizes for a specific electrical resistivity and for a specific relative permeability depending on each other and depending on the frequency of the primary alternating magnetic field. Solid and electrically conductive implanted devices show to generate less heating power compared to other types of implanted devices with equal dimensions, such as coil type implanted devices. Specific parameters for achieving maximum heating power are presented in this work along with the corresponding heating power for all types of implanted devices, except implanted devices with reactive power compensation. Moreover, with the results achieved in this publication, the ratio of generated heating power to required magnetic field strength can be maximized with respect to different limiting conditions, such as a specific operating frequency or a specific relative permeability of the implant material. Additionally, the SAR is shown to be below the limit, which is considered to not cause physiological stress for patients, for achieving an adequate rise of tissue temperature to conduct a thermal tumor ablation. This is very advantageous for the outcome of the tumor treatment and enhances the safety as well as the well-being of the patients significantly.