Two-Phase System for Generating a Higher-Frequency Rotating Magnetic Field Excited Causing Hyperthermic Effect in Magnetic Fluids

Abstract

This article presents a new method of excitation for a fast-changing rotating magnetic field (RMF) of higher frequencies (HF) causing the hyperthermic effect in magnetic fluids. The method proposed here uses a magnetic field exciter (inductor) consisting of a ferrite magnetic circuit and a system of coils connected in a two-phase arrangement. The proposed system is powered by two higher-frequency rectangular signals, with a 90-degree phase shift between each other, through HF transformers with ferrite cores. This paper presents the outcomes of the operation of RMFs in the frequency range of 38 kHz to 190 kHz, with a value of amplitude of magnetic field intensity H equal to 20 kA/m and increasing temperature, in a sample of APG513 magnetic liquid. The obtained results show that, in the range of the magnetic field intensities of moderate values, at a constant frequency f, the values of the time derivative of temperature are proportional to the square of the magnetic field intensity dT/dt~H². Moreover, the values of the temperature rate, which are measured with the constant value of the magnetic field intensity, are proportional to the square of the frequency dT/dt~f². At higher amplitudes of the RMF, the relationship dT/dt~H² is no longer fulfilled, and an inflexion point of this function appears. In the case of the highest values of the achieved intensity amplitudes (H = 20 kA/m), the parameter of the Langevin function achieves a value equal to ξ = 6.

1. Introduction

The application of magnetic hyperthermia in oncology has been utilised for many years, and it is mainly used as a therapy supporting radiotherapy and chemotherapy [1,2]. According to numerous reports in the literature [3,4,5,6,7], the effect of destroying cancer cells by means of magnetic hyperthermia consists of heating the tumour to a temperature of about 42–45 °C for a specific period, during which cancer cells are destroyed. This raised temperature exceeds a physiological value of 36.6 °C and the limited capability of the body’s regulatory system dissipates the heat. As a result, physiological changes appear in the tissues that allow chemotherapy or radiotherapy to interact more effectively. In addition, healthy cells have greater resistance to overheating and are, therefore, less sensitive to the thermal factor. A very important issue in the heating process is limiting the rise in temperature to the area of the tumour, i.e., where there are lesions.

Regarding the application of magnetic hyperthermia, an alternating magnetic field (AMF) is most often used to act on magnetic nanoparticles introduced previously into the tumour. The AMF excites these nanoparticles, which become a thermal energy source released into the surroundings [8,9,10,11]. In practice, magnetite is typically applied and its nanoparticles, covered with a layer of surfactant, constitute a poly-dispersed system [8]. The main mechanisms describing the effect of heat released to surroundings by nanoparticles in an alternating field are hysteresis and magnetic relaxation [3,8].

In the case of magnetic relaxation, the release of heat under the influence of an external, variable magnetic field can be based on the mechanisms of Brown and Néel. Brown’s mechanism [8] consists of rotating the magnetic moment vector with the entire magnetic particle (core) of the nanoparticle and is described by the expression ${\tau }_B=3{\eta }_S{V}_h/\left(k_{B}T\right)$, where V_h is the hydrodynamic volume of the particle. This mechanism occurs when the nanoparticle is in a liquid medium and is free to rotate. The Néel mechanism is based [3] on the rotation of the magnetic moment vector in relation to the core of the magnetic nanoparticle and is defined by the expression ${\tau }_N={\tau }_{o}ex{p}^{\left(\frac{\mathsf{\Delta }E}{k_{B}T}\right)}$ [3]. If both of these magnetising mechanisms take place simultaneously, the effective magnetising time is defined as ${\tau }_{eff}={\tau }_B·{\tau }_N/\left({\tau }_B+{\tau }_N\right)$ and the dominant mechanism is the one whose time is shorter. In slightly larger nanoparticles, there are also losses associated with magnetic hysteresis and the thermal power is then directly proportional to the frequency f of changes in the magnetic field and to the hysteresis loop area.

In the case when they are placed in a rotating magnetic field (RMF) and their revolutions follow the field without slipping, the hysteresis losses should disappear and the main loss mechanism should be related to the Brown rotation. However, when nanoparticles rotate with a lag behind the rotating field, the hysteresis losses also occur in proportion to the difference in field frequency and particle rotation.

Until recently, an oscillating magnetic field was used because it was easier to generate in conditions of relatively high signal frequency. However, the RMF can also be employed for this purpose. While the subject of the theoretical higher-frequency RMF is sufficiently present in scientific papers [12,13,14,15], the number of experimental works in this area is clearly limited [16,17]. The theoretical work by Raikher et al. [15] compared the influence of the rotating and alternating magnetic fields of higher frequencies on the value obtained by the calorimetric effect. The calorimetric effect in the rotating field was shown to be about 100% greater than in the oscillating field. Several other recently published experimental works [18,19,20,21,22] presented that, in the case of a rotating magnetic field, the heat release in iron oxide nanopowders was about three times greater than in the case of an oscillating field, despite the fact that the generating systems were based on the operation according to various concepts.

Among the works describing the RMF, are also those that concern the low-frequency range f < 1 kHz, where generation is performed by mechanical rotation of permanent magnets [23,24,25]. These solutions are not applicable in hyperthermia therapy.

The efficiency of the released heat energy in both processes is influenced by many factors, depending on the magnetic field parameters (frequency, amplitude of magnetic field intensity), the duration of exposure of nanoparticles to the field, as well as the type of magnetic material, surfactant layer thickness, grain size, or the type of liquid. Experiments conducted in many research centres have determined which of these conditions have a decisive impact on the reduction in magnetic material introduced into the body with the same thermal effects.

This work is devoted to the description of a new technical method of excitation: A rotating magnetic field, in which a magnetic system with ferrite cores has been powered by two electric rectangular waveforms with a phase shift of 90 angular degrees. This solution allows us to significantly reduce the cost of device construction, as there is no need to use expensive high-power amplifiers. Instead of these amplifiers, electronic keys, in the form of IGBT modules, were used.

2. Description of the System Generating a Rotating Magnetic Field

The proposed RMF-generating system, shown in Figure 1, includes the following components:

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Regulated DC voltage source;

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Two electronic switches (IGBT modules);

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L_SC_S series bandpass filter;

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Two HF transformers (TR) with ferrite cores;

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Magnetic torus with L_PC_P parallel circuits.

The system constructed to generate the RMF uses a regulated 500 W DC voltage source. This source is connected to electronic switches consisting of IGBT modules.

Both switches are mutually controlled by two rectangular signals out of phase by 90 angular degrees. As a result, rectangular waveforms appear at their outputs with amplitudes close to the voltage of the power source but shifted, in phase, by 90°. The signals from the outputs of the IGBT modules are connected to the serial branches of L_SC_S, which act as bandpass filters and are tuned to the fundamental harmonic of the square wave with frequency f. In order to obtain a higher value of RMF amplitude in the system, voltage step-up transformers (made from ferrite) were used.

The magnetic circuit system consists of a ferrite torus and poles with wound magnetising windings of inductance L_P, to which C_P capacitors are connected in parallel. The air-gap length in the centre of the torus is l = 27.7 mm.

An appropriate selection of resonant circuit elements (L_S, C_S, L_P, and C_P) makes their resonant frequencies equal. The remaining details describing the parallel magnetic circuits are included in earlier work [20]. The maximum value of the amplitude of the magnetic field intensity for the RMF in the constructed prototype reaches 20 kA/m, while the upper frequency range does not exceed 200 kHz. As can be easily demonstrated (see Appendix A), the maximum generated RMF amplitude depends on several factors. It is proportional to the voltage amplitude on the magnetising coil U₀ and inversely proportional to the frequency f, the turn number n₀, and the air-gap length l according to the relationship:

$$H ~\frac{U_0}{f{n}_{0}l}$$

(1)

The limitation of the upper frequencies results from the time needed to charge and discharge the parasitic capacities in the IGBT modules used. Then, at higher operating frequencies in the modules (i.e., above 150 kHz), a longer time is needed to reload the parasitic capacitors, which causes deformation of the shape of the voltage waveforms at the key outputs, as shown in Figure 2. Despite this, even when square waveforms at higher frequencies are strongly distorted, sinusoidal waveforms are still obtained on parallel LC resonant circuits. The circuits with coils and capacitors show the high-quality factor Q, i.e., the narrow passband Δf. Therefore, before calorimetric measurements, these circuits must be carefully tuned to the current value of the operating frequency.

It should be noted that the results of the calculation for the magnetic circuit of the RMF system, carried out in the initial stage of the design process with the use of analytical dependencies (Appendix A), has been confirmed in specially developed software for field analysis of the operating work states and to determine the magnetic field distributions. To analyse phenomena in the studied system, the finite element method (FEM) using the three-dimensional (3D) approach, based on the formulation of Ω − T₀ [19], was applied. Here, the nodal values of scalar potential Ω were used to define the distribution of the magnetic field, whereas to determine the values of the magnetomotive forces Θ (mmfs) as well as the phase currents i in the windings, the edge values of the vector electric potential T₀ were used [26,27]. Moreover, when formulating the RMF numerical model, the authors also took into account the relationships describing the transformation of magnetic field energy into heat energy in magnetic liquids. The view of the CAD model of the considered construction, with two phases of the RMF system and its prototype, is shown in Figure 3. A detailed description of the 3D field model and the obtained results of the design calculation are the subject of another (prepared) work and will not be discussed here.

In this work, the authors have limited themselves to the presentation of selected design calculation results, i.e., maps of the vector module of the magnetic field intensity H at half the height of the considered RMF system (Figure 4) and the spatial distributions of the radial component of the vector H on the surface S that represents the area of a sample (with the magnetic liquid) of about 12 mm in diameter for two selected situations in Figure 4. The specific situations are reported in Figure 4a,c (see also Figure 5). It is this component that best illustrates the distribution shape and variability of magnetic field intensity in the sample region. The distributions presented in Figure 4 refer to seven chosen time steps of different instantaneous values of the mmf Θ_k waveform in the kth phase, where k = 1, 2. The values of the phase mmf Θ_k have been calculated on the basis of the formula Θ_k = Θ_maxcos(ωt + (k − 1)·π/2), where Θ_max = 130 Ampere-turns. For the pth distribution, the value of ωt is ωt = (p − 1)·π/6, where p = 2 for the distributions in Figure 4b, etc. Moreover, when we look at these distributions, we can see the rotation of the considered magnetic field in the studied RMF system. It can also be observed that the module values of the vector H in the sample area are constants close to 6.8 kA/m. The calculations of the distributions, which have been presented in Figure 4 and Figure 5, were made for f = 190 kHz.

3. Calorimetric Results and Analysis of Experiment

This work presents calorimetric measurements in the RMF at five different frequencies: 38, 75, 116, 150, and 190 kHz. The tested sample of APG-513 magnetic fluid, 2 mL in volume, was put into a cooler connected to a thermostat that determined the initial measurement temperature. The temperature changes were observed using an optical sensor thermometer produced by FISO Technologies Inc. The properties of the applied magnetic fluid APG-513 are shown in

Table 1. Physical properties of APG-513 magnetic fluid.

Parameters
carrier fluid of MF	synthetic ester
magnetic core	magnetite
volume concentration	ϕV = 6.7%
dynamic viscosity (at 270C)	η = 150 mPa·s
saturation magnetisation of MF	Msat = 40 mT
mean size of magnetic core	<dm> = 16.2 nm
mean hydrodynamic NPs size	<dh> = 20.2 nm
mean magnetic volume of NPs	<Vm> = 2.26 × 10−24 m3
mean hydrodynamic volume of NPs	<Vh> = 4.32 × 10−24 m3
magnetic saturation of magnetite NPs	MS = 446 kA/m
mean magnetic moment of NPs	m = Vm·MS = 10.1 × 10−19 A·m2
Brownian relaxation time	τB = 472 μs
Néel relaxation time	τN = 385 ns

.

The measurement of the amplitude of the magnetic field intensity H for the RMF was based on the registration of the voltage induced in the probe’s coil with surface S_c at frequency f, which allowed one to calculate the value H using the basic laws of electromagnetic induction (i.e., Faraday’s law).

Figure 6a, Figure 7a, Figure 8a, Figure 9a and Figure 10a show the time records of temperature from t = 0 s up until the moment magnetic field was switched on (at t = 30 s), and during the RMF operation until the end-time of the recording (t = 150 s). The initial temperature course (t = 0–30 s) made it possible to assess whether the sample temperature was sufficiently stabilised before the RMF had been switched on. When the temperature fluctuations did not exceed 10 mK within 30 s, the magnetic field was switched on. Based on the time records of the temperature T(t), the value of the parameter dT/dt was determined, which is proportional to the loss of the thermal power P~dT/dt released in the sample.

Figure 6b, Figure 7b, Figure 8b, Figure 9b and Figure 10b show graphical dependencies of the temperature change rate on the amplitude of the magnetic field intensity H, plotted at different frequencies. In each case, it can be seen that the dT/dt function shows the inflexion point H₀ depending on the RMF amplitude which, for the lowest frequency f = 38 kHz, is H₀ = 8200 A·m⁻¹.

The position of this point along the H axis was determined by finding the second temperature derivative d²T/dt². The continuous line of the dT/dt function was obtained by fitting a third-degree polynomial to the measurement points.

Figure 11a shows the course of the Langevin function L(ξ, H) and the second temperature derivative d²T/dt² given in relation to the intensity amplitude of the RMF and the parameter ξ of this function. It can be seen that the inflexion point H₀ occurred when the parameter ξ ≌ 2.5, i.e., it was outside the linear region of the Langevin function. In many works [8,13,19,20,28], the dT/dt parameter is proportional to H² in a moderate range of intensities below the inflexion point H < H₀. At the same time, when the inequality (2πfτ)² << 1 holds, the dT/dt parameter is also proportional to f ². The commercial magnetic fluid that was used in our measurements fulfilled this condition, even at the highest frequency f.

Langevin’s considerations concern an assembly of atoms or molecules having permanent magnetic moments and not interacting with each other. Under the influence of an external magnetic field, such molecules try to align with their magnetic moments parallel to the field force lines. However, this alignment of moments is counteracted by thermal vibrations dependent on the temperature value. The magnetic moments of all molecules onto the direction of the magnetic field give the resultant magnetisation of the sample, which is described by the Langevin equation:

$$M\left(\mathsf{\xi }\right)=M_S\left(\coth \xi -\frac{1}{\xi }\right),$$

(2)

where M(ξ) is the magnetisation value depending on the parameter ξ and M_S is the magnetisation value in the magnetic saturation state.

The dimensionless parameter ξ of the Langevin function is determined from the expression:

$$\xi =\frac{{\mu }_{0}mH}{k_{B}T},$$

(3)

in which the magnetic permeability coefficient of vacuum μ₀ = 4π10⁻⁷ H/m, m [A·m²] is the magnetic moment of the molecule, T [K] is the absolute temperature, and k_B = 1.38 × 10⁻²³ [J/K]—the Boltzmann constant.

Langevin’s theory is often successfully used to describe the magnetisation of magnetic liquids with moderate concentrations of nanoparticles. Additionally, due to the poly-dispersion of nanoparticles, this fact is taken into account in the expression describing their magnetisation. In this case, the effective magnetisation of M_L(H) of a magnetic fluid is expressed as a superposition of many Langevin L(ξ) functions related to various fractions of colloidal nanoparticles [29] according to the formula:

$$M_L\left(H\right)=n\underset{0}{\overset{\infty }{{\displaystyle \int }}}m\left(x\right)L\left(\xi \right)f_p\left(x\right)dx,$$

(4)

where m and n are the magnetic moment and number density of particles, respectively, f_p(x) represents the function of distribution of particle size, and x is the diameter of a particle’s magnetic core.

The derivative of the Langevin function dL(ξ)/dξ = 1/3, for ξ = 0. In the case of the magnetic fluid APG-513, the numerical relationship between the parameter ξ of the Langevin function and the amplitude H is as follows: ξ = μ₀mH/k_BT = H/3333. This relationship was obtained for the mean magnetic moment of the nanoparticles <m> = 9.9 × 10⁻¹⁹A·m² and temperature T = 300 K. The tangent to the function L(ξ) at the point where ξ = 0 intersects the upper axis at H = 10 kA/m corresponds to ξ = 3.

Figure 11b shows the course of the tangent to the Langevin function for the range ξ = 0 ÷ 3 and its difference with the Langevin function. This difference is an indicator of the deviation of the initially linear course of the Langevin function with an increase in the amplitude value H. It transpires that 1% of this difference occurs when ξ = 0.78. This corresponds to a value of amplitude of the magnetic field H = 2.6 kA/m, while the 10% difference appears at ξ = 1.8, which corresponds to H = 6 kA/m.

When large amplitude values of the RMF intensity are used, especially those exceeding the inflexion point H₀, the power function of the dT/dt~Hⁿ type does not represent experimental values. At the same time, the Langevin function for ξ > 1 (i.e., H > 3300 A/m) clearly deviates from linearity. Then, the polynomial (H/a)ⁿ–(H/b)³, whose values fit well with the experimental points over the entire H range, can be used.

Figure 12 presents the relation of the dT/dt parameter as a function of the amplitude values of the magnetic field intensity for different values of frequencies. Equation (1) shows that the lower the frequency f, the greater the value of the maximum amplitude H_max is obtained. Such a relationship actually occurred in the experiment, where H_max = 20 kA/m was obtained at f = 38 kHz, whereas at f = 190 kHz, only H_max = 6.7 kA/m was obtained. The criterion [3,30] concerning the values of the magnetic field parameters used in magnetic hyperthermia (f and H) was closely met in each case if the inflexion point H₀ was not exceeded. The criterion [3,30] recommends that the product of the amplitude of the magnetic field intensity H and the frequency f should not be higher than 4.85 × 10⁸A·m⁻¹·s⁻¹. Some works in the literature states that this level can be exceeded to values of 5 × 10⁹A·m⁻¹·s⁻¹ [31,32,33] with no harm being caused to the patient. Thus, if the amplitude of the magnetic field intensity determined for the inflexion point H₀ is not exceeded, such a value can be tolerated by the organism.

The collective diagrams presented in Figure 12 are related to the limitation of the obtained changes in the value of the magnetising current intensity and the magnetic field at high frequencies. It is known that the reactance of the magnetic coils (X_L = 2πfL_p) increases with increasing frequency, which means that the magnetising current decreases. This, in turn, leads to a decrease in the H intensity. This explains the relatively high intensity obtained at low frequencies.

Table 2. Comparison of the products of amplitude of the magnetic field intensity H₀ and the frequency f with the value recommended by the criterion H₀·f = 4.85 × 10⁸A·m⁻¹·s⁻¹ [3,30]. The Langevin function parameter ξ₀ corresponds to the intensity of the RMF amplitude at the inflexion point H₀.

f [kHz]	H0 [kA/m]	ξ0 [-]	108·f·H0[A·m·s−1]
38	8.2	2.46	3.1 < 4.85
75	5.7	1.71	4.3 < 4.85
116	4.4	1.32	5.1 ≃ 4.85
150	3.3	0.99	4.9 ≃ 4.85
190	4.5	1.35	8.5 > 4.85

compares the products of the amplitude of the magnetic field intensity H₀ and the frequency f at the inflexion point of the function of dT/dt, with the value recommended by Brezovich for the criterion H₀·f = 4.85 × 10⁸A·m⁻¹·s⁻¹ [3,30]. Similarly, Figure 13 shows these relations for the frequencies selected in the experiment. The points marked (•) represent the RMF amplitude H₀ at which the inflexion point occurred. A slightly larger deviation occurs for extreme frequencies but, in these cases, another less restrictive criterion [31,32,33] was not exceeded.

Figure 14 shows dependences of the temperature increase rate dT/dt as a function of frequency for H = 1 kA/m, 2 kA/m, 3 kA/m, 4 kA/m, and H = 5 kA/m, i.e., dependences obtained from measurements. The measurement points were determined at frequencies f = 38, 75, 116, 150, and 190 kHz. In all of these cases, a power function of the $\frac{dT}{dt}={\left(\frac{f}{b}\right)}^k$ type was used, in which b and k were the fitting function parameters, which were obtained on the basis on the measurement data. It can be observed that the value of dT/dt depends strongly on the increase in the value of frequency f and the amplitude H.

In turn, the dT/dt waveforms presented in Figure 14 made for limited RMF values (H < 5 kA/m, ξ < 1.5) indicate that, in this range, the value of dT/dt is directly proportional to the function of power relative to the frequency: (dT/dt) = f^k, where k is the value of the exponent from fitting the function to the experimental points.

Finally, authors performed an additional experiment to compare the thermal effect for two kinds of magnetic fields, i.e., the rotation and alternating field. The temperature changes, presented in Figure 15, have been made for frequency f = 200 kHz and magnetic vector intensity H = 3.3 kA/m for both types of magnetic fields.

The upper run for the RMF was obtained with both amplifiers turned on. After turning on the magnetic field, the value of parameter dT/dt = 64 mK/s was obtained. When one of the amplifiers was turned off (AMF), the temperature was recorded with the temperature change rate dT/dt = 37.8 mK/s. The RMF amplitude (H = 3.3 kA/m) corresponds to the value of the Langevin function parameter ξ ≌ 1.

It is clearly visible that the use of two magnetic fluxes mutually shifted by 90° (i.e., RMF) also results in an increased rate of temperature rise by ca. 70% in relation to AMF, which was to be expected. Analogous results of investigation have also been obtained by authors of others works [16].

4. Conclusions

Within a moderate range of RMF intensity amplitudes, at a constant frequency f, the calorimetric effect is proportional to the square of the amplitude, i.e., dT/dt~H². In the case where the experiment was also performed under the condition of constant amplitude of the magnetic field intensity H, the temperature increase rate dT/dt was proportional to the square of the frequency, i.e., dT/dt~f². This conclusion is correct when (2πfτ) << 1.

At higher RMF amplitudes, an inflexion point is observed, above which the power relationship between dT/dt and the magnetic field intensity H is no longer satisfied.

The maximum value of the RMF amplitude obtained is influenced by various factors related to the design of the magnetic circuit, the air-gap length l, and the turn number of the magnetising coil n₀, as well as the power supply parameters (the adjustable voltage U_DC connected to the keys and the switching frequency f).