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Biocompatibility was evaluated by a CCK-8 assay. The inhibitory ratio of the treated group was Conclusion: These results demonstrate that it is possible to kill CSCs using targeted magnetic nanoparticles and an AMF and that magnetic fluid hyperthermia significantly inhibited the growth of grafted Cal tumors in mice. Keywords: magnetic nanoparticles, cancer stem cells, alternating magnetic field, tumor targeting. This work is published and licensed by Dove Medical Press Limited. By accessing the work you hereby accept the Terms. Non-commercial uses of the work are permitted without any further permission from Dove Medical Press Limited, provided the work is properly attributed.
For permission for commercial use of this work, please see paragraphs 4. The aim of this paper is to estimate the SAR of magnetite nanoparticles in biological media to quantitatively predict their heating efficiency in magnetic nanoparticle hyperthermia. In this respect we would like to stress that the behavior of an assembly of magnetic nanoparticles in viscous liquids and biological media is different [2,3].
It has been proved recently [13,24,25] that in biological media the magnetic nanoparticles can agglomerate within the biological cells or in the intracellular environment. The dense clusters of the nanoparticles turn out to be tightly bound to the surrounding media, so that the rotation of the nanoparticles as a whole is greatly hindered.
It is also important that the average distance between the centers of closest nanoparticles in the cluster is small, of the order of the particle diameter. Therefore, the strong magneto—dipole interaction within the clusters considerably affects the heating efficiency of the assembly [,24,25]. In addition, in the calculations performed, the fractal nature  of magnetic clusters in biological media is taken into account.
The numerical simulations are carried out using the Landau—Lifshitz LL stochastic equation [22,]. It is found that, similar to the case of interacting uniaxial nanoparticles  , the strong magneto—dipole interaction considerably decreases the SAR of fractal clusters of magnetite nanoparticles.
However, the dependence of the SAR on the mean nanoparticle diameter is retained, being less pronounced for strongly interacting nanoparticles. It is also important for clusters of magnetite nanoparticles with cubic and combined magnetic anisotropy that the maximal SAR values shift to larger particle diameters with respect to those for similar nanoparticles with uniaxial anisotropy . This is attributed to a decreased value of effective energy barriers for particles with cubic or combined anisotropy.
It has been previously shown for an assembly of uniaxial nanoparticles  that the existence of nonmagnetic shells of appreciable thickness at the nanoparticle surface considerably reduces the intensity of the magneto—dipole interaction within the cluster. A similar effect is also confirmed for clusters of nanoparticles with cubic or combined anisotropy. This effect can be used to improve the ability of magnetite nanoparticles to generate heat in an alternating external magnetic field. This shows the substantial potential of these nanoparticles for application in magnetic nanoparticle hyperthermia.
It has been recently shown  that the technique based on the stochastic LL equation is preferable for investigation of the properties of interacting assemblies of superparamagnetic nanoparticles with uniaxial anisotropy. In the present manuscript the same approach is used to study the behavior of dense clusters of magnetite nanoparticles with cubic or combined anisotropy.
The stochastic LL equation [22,] governs the dynamics of the unit magnetization vector of the i th single-domain nanoparticle of the cluster. The effective magnetic field acting on a separate nanoparticle can be calculated as a derivative of the total cluster energy.
For nanoparticles with cubic type magnetic anisotropy, the magneto-crystalline anisotropy energy is given by. Here K c is the cubic magnetic anisotropy constant, and e 1 i , e 2 i , e 3 i is a set of orthogonal unit vectors that determine an orientation of the i th nanoparticle of the cluster. For the shape anisotropy constant one obtains .
Next, the Zeeman energy of the cluster in an applied alternating magnetic field is given by. For nearly spherical, uniformly magnetized nanoparticles the magnetostatic energy of the cluster can be represented as the energy of the point interacting dipoles located at the particle centers r i within the cluster.
Then the energy of magneto—dipole interaction is.
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The thermal fields acting on various nanoparticles of the cluster are statistically independent, with the following statistical properties  of their components. It is well known [33,34] that the geometrical structure of a fractal cluster of nanoparticles is characterized by the relation where N p is the total number of the nanoparticles in the cluster, D f is the fractal dimension, and k f is the the fractal prefactor. The radius of gyration R g is defined as the mean square of the distances between the particle centers and the geometrical center of mass of the cluster.
Interestingly, in contrast to usual 3D clusters, the dimension D f of a fractal cluster is typically a noninteger number. Similar results were obtained also for clusters with other fractal descriptors.
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The inset shows an isolated magnetite nanoparticle of diameter D covered with a nonmagnetic shell of thickness t sh. Let us first consider the SAR of a dilute assembly of fractal clusters consisting of spherical magnetite nanoparticles. The clusters of nanoparticles are assumed to be tightly bound to surrounding media so that the nanoparticles cannot rotate as a whole under the influence of an alternating magnetic field. It is well known that magnetic nanoparticles are usually covered by thin nonmagnetic shells to protect them from oxidation . It was theoretically shown  that the intensity of the magneto—dipole interaction in dense clusters of uniaxial nanoparticles depends appreciably on the thickness of the nonmagnetic shell at the nanoparticle surface.
Figure 2: a Low frequency hysteresis loops of dilute clusters of spherical magnetite nanoparticles with cubi Therefore, for assemblies of interacting iron oxide nanoparticles with cubic anisotropy, the optimal particle diameter is considerably larger than that for an assembly of uniaxial nanoparticles .
This is because for a particle with cubic or combined magnetic anisotropy, the height of the energy barrier between various potential wells is much lower than that for a uniaxial nanoparticle of the same volume . By increasing the shell thickness one can decrease the intensity of the magneto—dipole interaction among the closest nanoparticles of the cluster.
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As a result, the area of the hysteresis loop increases as a function of t sh. A similar effect was also observed for interacting assemblies of nanoparticles with uniaxial magnetic anisotropy . Besides, the appreciable dependence of SAR on the nanoparticle diameter is revealed in all cases investigated. Evidently, due to the influence of the strong magneto—dipole interaction in fractal clusters, the SAR of the assembly of clusters decreases considerably with respect to that of an assembly of noninteracting nanoparticles.
Figure 3: SAR of dilute assemblies of fractal clusters of spherical magnetite nanoparticles with cubic anisot It is well known that the height of the effective energy barrier increases exponentially as a function of particle diameter. For small particle diameters, the effective energy barrier is too small at room temperature.
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Therefore, the alternating magnetic field has little influence on the assembly behavior. For large particle diameters the barriers are too large, so that the magnetization reversal of the particle is impossible or less probable under an alternating magnetic field of moderate amplitude. However, the probability for particle magnetization reversal increases with the increase in magnetic field amplitude . They are also qualitatively true for perfect magnetite nanoparticles of cubic external shape [8,10] having cubic magnetic anisotropy.
Actually, it follows from the Brown—Morrish theorem  that a single-domain nanoparticle of cubic shape has equal demagnetizing factors. Therefore, its magnetic properties in the first approximation are equivalent to that of a sphere. It is important to note, however, that the measurement of the angle dependence of the ferromagnetic resonance absorption indicates that the magnetite nanoparticles used in the experiment  possess not cubic, but combined or effective uniaxial anisotropy.
This behavior may be a consequence of random deviations of the nanoparticle shapes. It was recently shown  that for magnetic nanoparticles of soft magnetic type with cubic magneto-crystalline anisotropy, even relatively small perturbations of the spherical shape lead to an appreciable shape magnetic anisotropy. As a result, such nanoparticles possess combined or even effective uniaxial anisotropy. According to the Brown—Morrish theorem  , the magnetostatic properties of a single domain nanoparticle i.
For small shape deviations, the semiaxis ratio of an equivalent ellipsoid for simplicity, spheroid is close to unity. Thus the probability of the magnetization reversal at a fixed value of the magnetic field amplitude decreases. This effect deserves special consideration for assemblies of interacting magnetite nanoparticles.
Figure 4: Influence of particle elongation on the hysteresis loop shape for assemblies of noninteracting magn To study the influence of the shape anisotropy contribution on the assembly behavior, detailed calculations of the low frequency hysteresis loops have been carried out for dilute assemblies of fractal clusters of magnetite nanoparticles with small random spheroidal perturbations. Figure 5: a Evolution of the hysteresis loops of fractal clusters of magnetite nanoparticles with small aver Figure 6: SAR as a function of the transverse nanoparticle diameter D for dilute assemblies of fractal cluste