COLLOIDAL suspension of superparamagnetic (SP)

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1 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 4, APRIL Induction Heating of Magnetic Fluids for Hyperthermia Treatment Xufei Wang 1;2, Jintian Tang 2, and Liqun Shi 1 Institute of Modern Physics, Fudan University, Shanghai , China Department of Engineering Physics, Tsinghua University, Beijing , China The induction heating of magnetic fluids for magnetic induction hyperthermia treatments is theoretically analyzed, with regard to the influences by magnetic field parameters, material properties, and demagnetizing field effects in finite-size samples. A monodispersion model of noninteracting superparamagnetic particles, subjected to a magnetic field of the intensity ex 16 kam 1 and frequency 1 MHz, is used for the analysis. Calculation results show that the induction heating has a quasi-linear dependence on field intensity and a quasi-negative exponential dependence on field frequency. As for the influences by material parameters, respectively, the induction heating has a double-exponential-like dependence on the magnetic core size and a negative dependence on the coating layer thickness of the superparamagnetic particles. Similarly, the heating dependence on the carrier liquid viscosity is also a double-exponential-like relationship. Besides, the induction heating has a linear dependence on the volume fraction of the superparamagnetic particles, and a negative dependence on the demagnetizing factors, which are related to the sample shapes and orientations. Initial experiments are performed for validating the analytical calculation results. Index Terms Hyperthermia, induction heating, magnetic liquids, magnetic losses, superparamagnetic particles. I. INTRODUCTION COLLOIDAL suspension of superparamagnetic (SP) nanoparticles that are stably dispersed in a carrier liquid, also called magnetic fluid or ferrofluid, has been recently utilized for the magnetic induction hyperthermia (MIH) treatments to cancers [1] [8]. The principle is, when magnetic fluids are delivered to tumor sites and subjected to an ac magnetic field, the SP particles will produce damaging heat from ac magnetization loss, and kill the cancer cells at a controlled temperature of C. In order to reach the hyperthermia temperature with a minimum concentration in vivo, the induction heating ability of the magnetic fluids should be optimized. To test the induction heating ability for MIH, the magnetic fluids are usually cased as finite-size samples, and subjected to an ac magnetic field with the intensity kam and frequency MHz, which are typically used in clinical MIH treatments. However, the finite-sample-size effects on the heating performances of magnetic fluids are rarely concerned in related studies [7] [15]. To make clear the multiple influences and contributions on the heating performance of magnetic fluids, in this paper, a model analysis is made on the induction heating of finite-size samples of magnetic fluids, with regard to the influences by field parameters, material properties, and the demagnetizing field effects due to the sample shape and orientations. The paper is organized as follows. In Sections II and III, the general principles of magnetic relaxation loss and the effects of demagnetization field in finite-size magnetic samples are respectively discussed as the theoretical basis. In Section IV, to make sure the conditions of a uniform magnetization and a constant demagnetizing factor applicable for the instantaneous Manuscript received July 31, 2009; revised October 15, 2009; accepted October 26, First published December 15, 2009; current version published March 19, Corresponding author: X. Wang ( xufei.wang@gmail. com). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMAG magnetization in finite-size magnetic fluids, the propagating wavelength of magnetic field in the typical magnetic fluids for hyperthermia treatment is evaluated. In Section V, based on the general principles of magnetic relaxation loss and the physical mechanism of superparamagnetic relaxations, using a monodispersion model of noninteracting superparamagnetic nanoparticles, the induction heating of finite-size magnetic fluids is derived as a comprehensive expression, by which a detailed analysis is made on the dependences of the induction heating on magnetic field parameters, material properties and demagnetization field effects. In Section VI, initial experiment results are presented for validating the analytical conclusions. II. AC MAGNETIZATION LOSS In principle, a magnetic material subjected to ac magnetic field produces volumetric power dissipation, as given by is the permeability of free space, is the magnetic field intensity in the material, is the field frequency and is the imaginary component of complex susceptibility. For a motionless sample of magnetic fluids in a low-frequency field MHz, magnetic relaxation produces the main energy loss [11]. The magnetic relaxation process is described by is the magnetization intensity, is the relaxation time, is the equilibrium magnetization in the ac magnetic field, and is the equilibrium susceptibility [16], [17]. Substituting to (2) yields which gives the frequency-dependent complex susceptibility (1) (2) (3) (4) /$ IEEE

2 1044 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 4, APRIL 2010 from which the components of susceptibility are given by (5a) (5b) is the surface density of magnetic charges, and is the volume density of magnetic charges. Therefore, the demagnetizing field induced by the magnetic charges in the magnetized sample is expressed as Substituting (5b) to (1) yields a general expression of the magnetic relaxation loss (6) (13) In the condition of uniform magnetization,, surface magnetic charges are the only source of demagnetizing field, thus III. DEMAGNETIZING FIELD A finite-size magnetic sample in a constant magnetic field may induce a demagnetizing field, which reduces to the effective field in the sample, as described by is the magnetic susceptibility of the material. According to the principle of electrodynamics, magnetic field intensity can be expressed by magnetic potential as A magnetic dipole produces the magnetic potential as (7) (8) (9) (14) (15) (10) is the radius vector from the dipole center to field point. Taking the magnetized sample as an assembly of magnetic dipoles, the total magnetic potential induced by the sample is (11) is the magnetization intensity in the sample. Differential quotient of field point gives the inverse value of the quotient of source point, thus, and (12) is defined as demagnetizing tensor. Set the major axes of the sample as coordinate axes, then when,, the demagnetizing tensor reduces to (16) (17) Constants are called the demagnetizing factors, which are only dependent on the sample shape. The knowledge of the demagnetizing factors of magnetized shapes enables us to evaluate magnetostatic energies whenever uniform or quasi-uniform states are in play [18] [20]. The calculations by demagnetizing factors are exact for the shapes bounded by a quadratic surface (e.g., ellipsoids, paraboloids or hyperboloids), and approximate for other shapes that do not support perfectly uniform magnetizations, such as cylinders or circular plates. For example, when a uniform field is applied along the longitudinal axis of a cylinder sample (radius, length ), the magnetization in the sample is quasi-uniform along the axis. Set the coordinate axes along the cylinder axes, the transversal magnetizations are (18)

3 WANG et al.: INDUCTION HEATING OF MAGNETIC FLUIDS FOR HYPERTHERMIA TREATMENT 1045 and the induced demagnetizing field is uniform along the longitudinal axis as (19) According to (15), the demagnetizing factor along the longitudinal axis is derived as IV. PROPAGATING FIELD For the effective field propagating in the sample, the propagation factor is, is the wavelength and is the propagation distance in the material. Therefore, when the sample size, the instantaneous magnetization in the sample can be calculated as uniform. The propagating wavelength of the field is given by (25) is the corresponding wavelength in free space, is the refractive index, is the relative permittivity, and is the relative permeability of the sample material. For a magnetic fluid of low concentration, initial susceptibility is derived by Langevin relationship [21], [22] (20) and the initial permeability is (26) Due to the symmetry, the transversal demagnetizing factors is also determined as (21) Therefore, the longitudinal and transversal demagnetizing factors of the cylinder are (22a) (22b) From (7), (8), and (19), for the condition of uniform magnetization, we have (23a) (23b) is the demagnetizing factor related to the sample shape and orientations. When the finite-size sample is homogeneously magnetized by an external ac magnetic field, replacing in (23) with the complex susceptibility yields (24a) (24b) is the demagnetizing field and the effective field in the sample. Note that (24a) and (24b) only applies to the case of uniform magnetization. (27) is the saturation magnetic moment, the magnetic volume, and the solid volume fraction of the SP particles, is Boltzmann constant, and (K) is temperature. In a low magnetic field, e.g.,, approximately, one can take, then (28) As for the permittivity, in low concentration magnetic fluids, it can be taken as having the same order of magnitude of the carrier liquid. With the approximations, one can estimate the wavelength of the propagating field. For example, for a waterbased magnetite ferrofluid,,, and, the wavelength is calculated around. Therefore, for a frequency, the sample size below 1 m for table-top experiments is small enough to support a uniform instantaneous magnetization. V. CALCULATION RESULTS It is known that in induction heating, a solid magnetic work piece may display a shape-and-orientation-dependent heating performance, due to the different contributions from hysteresis and eddy current losses [23] [25]. The same question also arises for magnetic fluids, in which magnetic relaxations produce the energy loss. Many studies [7] [15] have been made on the induction heating of SP particles. However, finite-sample-size effects on the induction heating are rarely concerned. Actually, for magnetic fluids, demagnetizing field may also be induced in finite size samples. According to (24b), the demagnetizing field is dependent on external field, material parameters, and sample shape and orientations. Substituting (24a) into (1) yields a general expression of the AC magnetization loss with demagnetizing field effects (29)

4 1046 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 4, APRIL 2010 TABLE I TYPICAL MATERIAL PARAMETERS OF A LOW CONCENTRATION MAGNETITE FERROFLUID (TEMPERATURE T = 300 K) Substituting (5) into (29), one gets the expression of relaxation loss for finite-size samples with demagnetizing field effects (36) (30) The induction heating of magnetic fluids is mainly from the Brownian-Néel relaxation loss of SP particles [17], [26] [34]. In Brownian relaxation, the magnetic moment of SP particle is locked to the crystal axis and aligns with external field. When subjected to an alternating magnetic field, the particle moment rotates as well. In Néel relaxation, the magnetic moment rotates within the crystal with the applied magnetic field. The Brownian relaxation time is given by and the Néel relaxation time is given by (31a) (31b) is the viscosity coefficient of the carrier liquid, is the hydrodynamic volume and the anisotropy constant of the particles. A value of is used [22]. In a SP particle system, Brownian and Néel processes take place in parallel, and the effective relaxation time is given by (32) From the Langevin relationship of superparamagnetism [21], [22], the equilibrium susceptibility is given by (33) is saturation magnetization of the particle system. Substituting (5) and (31) (33) into (24a) yields an implicit relationship between and, as given by (34) Replacing and in (34) with and, is the radius of magnetic core and is the coating layer thickness of SP particles, one gets (35) Substituting (31) (33) into (30), one gets the expression of the Brownian-Néel relaxation loss in finite-size sample magnetic fluids, as given by (37) However, from (35) it is still difficult to derive an explicit relationship of the effective field on the external field and the material parameters. Thus an explicit function of on the known parameters is also unavailable from (37), in which the effective field remains an implicit term. In following, numerical calculations will be performed with (35) and (37), firstly on the field-parameter dependence of the induction heating, and besides, the unary dependences on the material parameters,,,, and, respectively. The typical values and ranges of the material parameters of a magnetite ferrofluid (Table I) are used for the calculations. The typical field intensity is, with the typical value 8 ; and the typical field frequency is 1000 khz, with the typical value 100 khz. The typical temperature is. Though the material parameters in real magnetic fluids are usually correlated [35], to date there is no widely accepted theory for an accurate prediction of the relationships. For the low-concentration magnetite ferrofluids, the parameters can be assumed uncorrelated. A. Field-Parameter Dependences The field-parameter dependence of the induction heating is numerically calculated for the typical material parameters. The calculations are confined to and. From the results (Fig. 1), it can be seen that the induction heating has a quasi-linear dependence on the field intensity and a quasinegative exponential dependence on the field frequency. It appears that the quasi-linear dependence on field intensity is contradictory to (37), the field intensity appears squared. Actually, the effective field intensity in the sample is related with the external field intensity, as indicated by the implicit relationship in (35). For the typical field parameters of and and the typical material parameters in Table I, the implicit relationship gives quasi-linear curves close to, as plotted in Fig. 1(d). If the demagnetizing field is neglected and the effective field is assumed equal to the external field, substituting (31) (33) into (6) and replacing the effective field with

5 WANG et al.: INDUCTION HEATING OF MAGNETIC FLUIDS FOR HYPERTHERMIA TREATMENT 1047 of the induction heating on field parameters are basically same, i.e., a quasi-linear dependence on magnetic field intensity and a quasi-negative exponential dependence on field frequency. B. Material-Parameter Dependences 1) Magnetic Core Size: First of all, a suitable range of for the calculations should be considered. According to the relaxation mechanisms of SP particles shown in (31) and (32), for a measurement time, to exhibit an intrinsic superparamagnetsim of Néel relaxation, the time constant should satisfy (40) from which the critical radius of intrinsic superparamagnetism for the magnetic cores is given by (41) Similarly, to exhibit an extrinsic superparamagnetism of Brownian relaxation, the rotational diffusion time of the particles should satisfy (42) Fig. 1. Field-parameter dependence of the induction heating: (a) binary dependence on field intensity and frequency, (b) quasi-linear dependence on field intensity, and (c) quasi-negative exponential dependence on field frequency. (d) Comparison between the external field intensity and the effective field intensity in the magnetic fluids of the typical material parameters. external field, one can get an approximate formula of the heating power as from which the critical radius of extrinsic superparamagnetism for the particle size is given by (43) Note that refers to the blocking size of hydrodynamic volume that includes both the magnetic core and the non-magnetic coating layer of the particles; as refers to the blocking size of the magnetic cores. For a monodispersion exhibiting both the relaxations, the particle size should satisfy (38) and (44) (39) It can be seen that for a constant frequency, the dependence of on is a quasi-linear relationship because the Langevin function approaches to 1 for an increasing value of ; as for a constant field, for an increasing frequency, the heating power approaches to a constant value, which is determined by the constant magnetic field intensity and the material parameters. Less contribution can be made by increasing the field frequency on the induction heating power. Because the real effective field in the magnetic fluid is close to the external field for the material parameters, as shown in Fig. 1(d), it can be deduced that the real dependences (45) at the same time. According to (44) and (45), for a magnetite ferrofluid of and, when and, one gets, and. It can be seen that the typical parameters ranges of and in Table I are small enough to ensure both the states of superparamagnetism. Dependence of the induction heating on magnetic core size is calculated in the range of 20 nm, and respectively, for different parametric values of the other parameters. The results in Fig. 2 show that the induction heating has a double-exponential-like dependence on the magnetic core size, with a peak heating appearing at the magnetic core size of 10 nm. In Fig. 2(b), a strange curve peak is detected for the frequency above 200 khz, corresponding to a relatively lower value of and shaper peak of higher heating power in comparison

6 1048 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 4, APRIL 2010 Fig. 2. Double-exponential-like dependence of the induction heating on magnetic core size, respectively for different parametrical values of (a) field intensity, (b) field frequency, (c) coating layer thickness, (d) carrier liquid viscosity, (e) solid volume fraction, and (f) demagnetizing factor. Fig. 3. Negative dependence of the induction heating on coating layer thickness, respectively for different parametrical values of (a) field intensity, (b) field frequency, (c) magnetic core size, (d) carrier liquid viscosity, (e) solid volume fraction, and (f) demagnetizing factor N. with the frequencies below 200 khz. This behavior is due to the mathematical property of the heating power dependence on magnetic core size and field frequency. By the approximation in (38), the dependent curve of on for different is also evaluated, the analytical results show the same behavior, and for the magnetic fluid in Table I, the value of corresponding to the peak heating can even be analytically calculated as for and ; for and ; for and ; for and ; for and, and for and. The results indicate that the value of for peak heating is negatively dependent on both the field intensity and the field frequency. 2) Coating Layer Thickness: The coating layer effects on the induction heating of the magnetite ferrofluid are calculated, for the thickness 4 nm, and respectively, for different parametric values of the other parameters. From the results in Fig. 3, it can be seen that the non-magnetic coating layer of SP particles basically tends to lead to a reduction of the heating performance of the ferrofluids. 3) Solid Volume Fraction: Induction heating of the ferrofluid simply display a linear dependence on the volume fraction of the solid SP particles, as shown in Fig. 4, 0.07 [11]. 4) Carrier Liquid Viscosity: For magnetic fluids with constant particle size, according to (45), to ensure an extrinsic superparamagnetism, the carrier liquid viscosity should satisfy, the crit-, which is far For the typical values in Table I, when ical viscosity of the carrier liquid is (46)

7 WANG et al.: INDUCTION HEATING OF MAGNETIC FLUIDS FOR HYPERTHERMIA TREATMENT 1049 Fig. 4. Linear dependence of the induction heating on solid volume fraction, for different values of (a) field intensity and (b) frequency. larger than the values of real magnetic fluids. Here the calculations on the viscosity effects are performed for, and respectively, for different parametric values of the other parameters. From the results (Fig. 5), one can see the induction heating also has a double-exponential-like dependence on the carrier liquid viscosity, with the peak heating around the region of to. Besides, the temperature dependence of liquid viscosity [36] may also influence the heating performance. For example, in a water-based ferrofluid, when temperature rises from 300 K to 373 K, the viscosity of water decreases from to. According to the heating dependence on carrier liquid viscosity, from Fig. 5, it can be seen that the heating may be enhanced due to the temperature rising. C. Demagnetizing Field Effects Firstly, the demagnetizing field effects on the induction heating of finite-size sample magnetic fluids are calculated on the demagnetizing factor, for, and respectively, for different parametric values of the other parameters. From the results in Fig. 6, it can be seen that the induction heating has a negative dependence on the demagnetizing factor. The dependence of the induction heating on demagnetizing factors can be further described by specific sample shape and orientations. For example, for a cylindrical shape, according to (22), the demagnetizing field effects can be further calculated with regard to the aspect ratio and orientations. The results are shown in Fig. 7. VI. EXPERIMENT RESULTS To validate the analytical calculation results, initial testing experiments are performed on the induction heating of a water based magnetic fluid of particles (Fig. 8(a)), which have a mean radius of 10 nm of the magnetic core and a concentration of mg/ml. The magnetic fluid is cylindrically cased in an adiabatic tube and subjected to an alternating magnetic field, which is generated in the center of an induction coil (Fig. 8(b)) driven by a medium-frequency power supply for induction heating (SPG-40, Shuangping Power Supply Technology Co. Ltd.). According to the theoretical frequency dependence in Fig. 5. Double-exponential-like dependence of the induction heating on carrier liquid viscosity, respectively for different parametrical values of (a) magnetic field intensity, (b) frequency, (c) magnetic core size, (d) coating layer thickness, (e) solid volume fraction, and (f) demagnetizing factor N. Fig. 1(c), a magnetic field intensity of 4.8 ka/m and a frequency of 180 khz are used in the experiments. Temperature-increasing curves of the magnetic fluid are measured. The results show that the heating performance of the finite-size sample has distinct orientation dependence (Fig. 8(c)), which corresponds to the analytical prediction of the demagnetizing field effects. The influence of particle concentration is also tested, with the results shown in Fig. 8(d). It can be seen that the heating ability is enhanced by higher concentration of the particles, with the same influence by the orientations in the magnetic field. Further experiments for validating the calculations on field parameters, particle sizes and viscosities are still ongoing. The complete experimental work will be finished and composed for a future report paper.

8 1050 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 4, APRIL 2010 Fig. 7. Dependence of the induction heating on the aspect ratio a = h=r (length h and radius R ) of a cylindrical sample, for different values of (a) field intensity, with f = 100 khz, and (b) field frequency, with h =8kAm. Fig. 6. Negative dependence of the induction heating on demagnetizing factor of finite-size samples, respectively for different parametrical values of (a) field intensity, (b) field frequency, (c) magnetic core size, (d) coating layer thickness, (e) carrier liquid viscosity, and (f) solid volume fraction. Fig. 8. Experimental validation of the calculated heating performance: (a) SEM image of the Fe O nanoparticles in the sample magnetic fluid, (b) induction heating setup of a cylindrically cased magnetic fluid, with the magnetic field of 4.8 ka/m,180 khz (corresponding to a current level of 200 A), and the adiabatic tube of 10 cm length and 1 cm diameter, with different orientations in the field, (c) temperature increasing curve of the magnetic fluid for different orientations in the field, (d) temperature increasing curve of the magnetic fluid with different orientations and different particle concentrations. VII. CONCLUSION Based on a collection of the general principles for a superparamagnetic nanoparticle system, a model analysis is made on the induction heating of the finite-size samples of magnetic fluids. Dependences of the induction heating on field parameters, material properties and sample-shape-related demagnetizing field effects are numerically calculated, leading to the following conclusions: Induction heating of the finite-size sample magnetic fluids has a quasi-linear dependence on magnetic field intensity, and a quasi-negative exponential dependence on field frequency. Therefore, in the MIH treatments by magnetic fluids, higher fields may improve the heating effects. However, from the quasi-negative exponential relation of the frequency dependence, a proper frequency can be practically selected. For example, for the heating dependence shown in Fig. 3(c), a frequency around 100 khz is practically adequate for MIH. Besides, the maximal field intensity is also practically limited, due to the security considerations and engineering difficulties. For constant field parameters, the magnetic fluid reaches a peak heating at a specific size of the magnetic cores. However, the non-magnetic coating of the SP particles tends to lead to a reduction of the heating performance. Therefore, to improve the heating ability of magnetic fluids, magnetic core size and coating layer thickness of the SP particles need to be carefully controlled during preparations. Similarly, the heating dependence on carrier liquid viscosity also has a double-exponential-like relationship. Considering the real viscosities of the body fluids (e.g., blood, tissue fluids or lymph fluids) around cancer tissues, the material parameters of the SP particles should be well designed, thus to improve the heating effects of the magnetic fluids in MIH treatments.

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New York: McGraw-Hill, Xufei Wang was born in Taiyuan, China, in He studied electronic engineering at Hangzhou Dianzi University, Hangzhou, China, and received the Diploma degree in From 1999 to 2005, he studied in the Institute of Plasma Physics in Chinese Academy of Sciences, Hefei, China, and received the Master degree in applied nuclear technology in 2002, and then the PhD. degree in nuclear science and engineering in From October 2005 to October 2006, he worked in the National Institute of Radiological Sciences, Chiba, Japan, as a Postdoctoral Research Fellow on the development and applications of accelerator-based single-ion microbeams for biological and material researches. From October 2006 to March 2009, he worked in Tsinghua University, Beijing, China as a Postdoctoral Researcher on the material researches and facility developments of magnetic induction hyperthermia for cancer treatments. In April 2009, he joined the Institute of Modern Physics in Fudan University, Shanghai, China, mainly working on the single-ion microbeam for medical and material applications. His main interests include the applications of single-ion microbeam for nanomaterial and biology researches, and further researches of the nanomagnetism and facilities developments for magnetic hyperthermia treatment. Dr. Wang is a member of the Chinese Society of Nanoscience and Technology, China.