# Application of Magnetosomes in Magnetic Hyperthermia

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{0}= 50–100 Oe. The maximum SAR value only weakly depends on the diameter of the nanoparticles and the length of the chain. However, a significant decrease in SAR occurs in a dense chain assembly due to the strong magneto-dipole interaction of nanoparticles of different chains.

## 1. Introduction

_{0}of the order of 100–200 Oe. It was proved [15] that the effect of an AC magnetic field on living organisms is relatively safe under the condition H

_{0}f ≤ 5 × 10

^{8}A(ms)

^{−1}, though recently safety limit was increased [16] up to H

_{0}f ≤ 5 × 10

^{9}A(ms)

^{−1}. In addition, only mildly toxic and biodegradable nanoparticles can be used in magnetic hyperthermia to reduce the risk of possible side effects. Based on these requirements, magnetic nanoparticles of iron oxides are considered to be the most suitable for application in magnetic hyperthermia [1,2,3,11,12,16,17,18,19].

_{2}O

_{3}[42] or magnetite, Fe

_{3}O

_{4}[45].

_{0}= 126 Oe. High SAR values in magnetosome assemblies were also obtained in a number of subsequent experimental studies [5,25,39,45]. It is known [37,38,43] that various types of magnetotactic bacteria synthesize nanoparticles with different characteristic sizes, from 20 to 50 nm. In the bacteria the nanoparticles are arranged in the form of long chains consisting of 6–30 particles of approximately the same diameter. Existing experimental techniques make it possible to isolate magnetosomes from bacteria both in the form of single particles and in the form of chains with different numbers of nanoparticles in the chain [25,37,38,39,40,41,42,43].

_{0}= 50−100 Oe. However, it is found that the SAR of the chain assembly decreases significantly with an increase in its average density. It seems that the results obtained will be useful for the optimal choice of the geometric parameters of magnetosome chains to further increase their heating ability for successful application in magnetic hyperthermia.

## 2. Model and Methods

_{1}= γ/(1 + κ

^{2}), ${\overrightarrow{H}}_{ef,i}$ is the effective magnetic field, ${\overrightarrow{H}}_{th,i}$ is the thermal field, and N

_{p}is the number of nanoparticles in the chain. The effective magnetic field acting on a separate nanoparticle can be calculated as a derivative of the total chain energy

_{a}+ W

_{Z}+ W

_{m}is a sum of the magnetic anisotropy energy W

_{a}, Zeeman energy W

_{Z}of the particles in applied magnetic field, and the energy of mutual magneto-dipole interaction of the particles W

_{m}. Since magnetosomes released from bacteria are coated with thin non-magnetic shells, there is no direct contact between the magnetic cores of the nanoparticles. Therefore, the exchange interaction of neighboring nanoparticles in the chain can be neglected.

_{B}is the Boltzmann constant, δ

_{α}

_{β}is the Kroneker symbol, and δ(t) is the delta function. The numerical simulation procedure is described in details in [54,55].

## 3. Results

_{0}= 50–150 Oe, since the use of AC magnetic fields of small amplitude is preferable in a medical clinic.

#### 3.1. Dilute Chain Assembly

_{p}, and on the orientation of the external magnetic field with respect to the chain axis. For completeness, the chains with different average distances a between the centers of the nanoparticles are considered. Note that all geometric parameters mentioned can be adopted properly during the experimental design of a chain from individual magnetosomes of approximately the same diameter [5,7,40,41,42]. In particular, the distance a between the centers of successive nanoparticles in a chain is determined by the thickness of non-magnetic shells on their surfaces. The latter protect the nanoparticles from the aggressive action of the medium. It is worth noting that, since the cubic magnetic anisotropy constant of magnetite is negative, K

_{c}= −1 × 10

^{5}erg/cm

^{3}, the quasi-spherical magnetite nanoparticles have eight equivalent directions of easy anisotropy axes [53]. In the calculations performed it is assumed that the easy anisotropy axes of individual magnetosomes are randomly oriented, since it is hardly possible to make multiple easy anisotropy axes of various magnetosomes parallel when a chain is created. Saturation magnetization of the magnetosomes is assumed to be M

_{s}= 450 emu/cm

^{3}[25,45]. It is also supposed that the magnetosome chain cannot rotate as a whole, being distributed in a medium with a sufficiently high viscosity or being tightly bound to surrounding tissue.

_{p}= 30, AC magnetic field frequency f = 300 kHz, field amplitude H

_{0}= 50 Oe. The AC magnetic field is applied parallel to the chain axis, the easy anisotropy axes of various particles in the chain are randomly oriented. The calculation results are averaged over a sufficiently large number of independent chain realizations, N

_{exp}= 40–60. The temperature of the system is T = 300 K.

_{0}. With increasing H

_{0}the maximum of the assembly SAR grows, and the position of the maximum falls at a shorter distance between the particles of the chain. Indeed, in Figure 2b the maximum values of SAR = 440.1, 854.6, and 1281.0 W/g are observed at a/D = 1.5, 1.35, and 1.2 for the magnetic field amplitudes H

_{0}= 50, 100, and 150 Oe, respectively. In general, as Figure 1 and Figure 2 show, for a dilute oriented assembly of magnetosome chains with an optimal choice of the a/D ratio one can obtain rather high SAR values, of the order of 400–450 W/g already in an AC magnetic field with a relatively small amplitude H

_{0}= 50 Oe.

_{0}, shown in Figure 1a and Figure 2b, is explained by the influence of a strong interacting field H

_{d}acting between closely located particles of the chain. Figure 3a shows the instantaneous distribution of interacting fields for an individual chain consisting of N

_{p}= 30 nanoparticles of diameter D = 20 nm at a time when the AC magnetic field is close to zero. The z axis is assumed to be parallel to the axis of the chain, the amplitude and frequency of the AC magnetic field are H

_{0}= 50 Oe, and f = 300 kHz, respectively. As Figure 3a shows, in the central part of the chain, due to the summation of the magnetic fields of individual nanoparticles, the longitudinal component of the interacting field reaches sufficiently large values, H

_{dz}= 250 Oe, significantly exceeding the amplitude of the AC magnetic field, H

_{0}= 50 Oe.

_{dx}|, |H

_{dy}| < 50 Oe. Obviously, the irregular distribution of the interacting field on individual particles, shown in Figure 3a is associated with a random orientation of the easy anisotropy axes of individual nanoparticles.

_{sf}of the chain is determined by the conditions near its ends. Calculations show that for H

_{sf}< H

_{0}the chain magnetization reverses as a whole in a process similar to the giant Barkhausen jump in iron-rich amorphous ferromagnetic microwires [56]. The switching field H

_{sf}of the chain increases with decreasing average distance between the particles in the chain, since the intensity of the magnetic dipole interaction increases if neighboring nanoparticles are located closer to each other. When the distance between the particle centers decreases, the switching field of the chain can reach values exceeding the amplitude of AC magnetic field, H

_{sf}> H

_{0}. Under this condition, the magnetization reversal of the chain is impossible. As Figure 1b shows, when the reduced distance between the particle centers decreases from a/D = 1.85 to a/D = 1.7, the area of the assembly hysteresis loop reduces sharply. This leads to a sharp drop in the SAR of the corresponding assembly in Figure 1a. However, the drop in the hysteresis loop area in Figure 1b does not occur immediately to zero, due to fluctuations in the switching fields H

_{sf}of individual chains of the assembly. As a result, even at a/D = 1.7 some of the assembly chains are still able to reverse their magnetizations in applied magnetic field. This, however, leads to a sharp narrowing of the vertical size of the low frequency hysteresis loop of the whole assembly. With a further decrease of the a/D ratio, the fraction of chains capable of magnetization reversal at given magnetic field amplitude H

_{0}tends to zero. As a result, the ability of the assembly to absorb the energy of an AC magnetic field disappears.

_{0}the magnetization reversal of the chain is possible at smaller reduced distances a/D. This leads to a shift in the positions of the SAR maximum to smaller reduced distances and to increase in the maximum SAR values.

_{0}. For ratios smaller than optimal, a/D < (a/D)

_{0}, the vertical size of the hysteresis loop is considerably decreased, M/M

_{s}< 1. On the other hand, for ratios larger than optimal, a/D > (a/D)

_{0}, the width of the assembly hysteresis loop is reduced, 2H

_{max}< 2H

_{0}.

_{p}= 4–6. For example, Figure 3b shows the dependences of SAR on the number of particles, N

_{p}= 2–30, in the chain assemblies with particles of various diameters, D = 20–40 nm. The calculations presented in Figure 3b are performed at optimal ratios a/D = 1.5, 1.85, and 2.2 for particles with diameters D = 20, 30, and 40 nm, respectively. These optimal a/D values were determined previously at frequency f = 300 kHz and amplitude H

_{0}= 50 Oe. According to Figure 3b, regardless of the nanoparticle diameter, a sharp increase in SAR as a function of the number of particles in the chains occurs in the range N

_{p}≤ 4–5. However, with a further increase in the chain length, the SAR values of the assemblies change slowly.

_{0}= 50 Oe, the number of particles in the chains being N

_{p}= 30. The reduced distances between the centers of neighboring nanoparticles in the chains are chosen optimal, so that a/D = 1.5, 1.85, and 2.2 for particles with diameters D = 20, 30, and 40 nm, respectively.

#### 3.2. Interaction of Magnetosome Chains

_{cl}= 280 nm, and a height L

_{cl}= 480 nm. The axis of the chains is parallel to the axis of the cylinder, but the positions of the chains in the cluster are distributed randomly, as shown schematically in Figure 5b. The AC magnetic field is applied along the chain axis. In the calculations performed it is assumed that the centers of the nanoparticles in the chains are located at an optimal distance, a = 2.2D, for nanoparticles with diameter D = 40 nm. This allows one to compare the numerical results with the data shown in Figure 1a for a dilute assembly of individual chains with the same nanoparticle diameter. The frequency and amplitude of the AC magnetic field are f = 300 kHz and H

_{0}= 50 Oe, respectively.

_{p}= 6, but the number of chains N

_{ch}in a cluster of given diameter D

_{cl}= 280 nm varied from 4 to 10. In this manner we can change the total number of nanoparticles in the cluster, N

_{p}N

_{ch}, as well as the cluster filling density, η = N

_{p}N

_{ch}V/V

_{cl}, where V = πD

^{3}/6 is the volume of the nanoparticle, ${V}_{cl}=\pi {D}_{cl}^{2}{L}_{cl}/4$ being the volume of the cylindrical cluster. SAR calculations of dilute assemblies of clusters with different numbers of chains N

_{ch}are averaged over a sufficiently large number of independent realizations of random clusters, N

_{exp}= 40–60.

_{ch}= 4, 6, 8, and 10, respectively. It is found that with increase in the cluster filling density, the SAR of the assembly of clusters rapidly decreases as follows: SAR = 270.9, 145.5, 98.2, and 62.5 W/g. It should be noted that for a dilute assembly of non-interacting chains (see Figure 1a) with the same chain geometry, frequency and amplitude of the AC magnetic field, the SAR of the assembly at maximum reaches the value 454 W/g for the particles with optimal ratio a/D = 2.2. Thus, a significant drop in the SAR assembly due to the magnetic dipole interaction of individual chains must be taken into account when analyzing experimental data.

## 4. Discussion

_{0}= 310 Oe. In agarose gel the SAR of the assembly at the same frequency and magnetic field amplitude turns out to be somewhat lower, SAR = 635 W/g, since in a medium with increased viscosity, the Brownian contribution to SAR is significantly reduced. These data are in agreement with the high SAR values obtained previously for magnetosome assemblies by Hergt et al. [37].

_{0}= 380 Oe. Nevertheless, it seems hardly possible to uniformly distribute and correctly orient within the tumor large bacteria of 2–5 μm in length. This can probably be done much easier using short chains of magnetosomes having 4–6 nanoparticles in length. For example, the optimal length of a chain consisting of five nanoparticles with diameter D = 20 nm is given by only 140 nm. In addition, when working with chains created from individual magnetosomes, it becomes possible to optimize the geometry of the chains, that is, to ensure the optimal a/D ratio, matching it with the applied value of H

_{0}.

_{p}= 4–6, and introduced into the tumor for magnetic hyperthermia. A typical concentration of magnetic nanoparticles introduced into a tumor is given by 25 μg in iron of nanoparticles per mm

^{3}of tumor [7,41,42]. Taking into account the density of magnetosomes around 5 g/cm

^{3}, and assuming nearly uniform distribution of the magnetosome chains within the tumor, one obtains the average density of the assembly η = 0.005. Such chain assembly can be considered quite diluted, so that the results obtained in Section 3.1 above can be applied for the theoretical estimation of the assembly SAR.

_{Fe}at frequency f = 198 kHz. As a result, in order to obtain the required tumor temperature of 43–46 °C, it was necessary to use rather large amplitudes of the AC magnetic field, H

_{0}= 110–310 Oe. Similarly, only relatively small values of SAR = 89–196 W/g

_{Fe}were obtained in [40] at frequency f = 198 kHz at sufficiently large magnetic field amplitudes, H

_{0}= 340–470 Oe. This may indicate that the geometric structure of the magnetosome chains, in particular, the a/D ratio, was not optimal in these experiments. Indeed, the individual magnetosomes used to construct the magnetosome chains [42] were covered by rather thin poly-L-lysine shells with a thickness of t = 4–17 nm. Therefore, for a significant fraction of the magnetosome chains with an average particle diameter D = 40.5 ± 8.5 nm, the ratio a/D = 1 + 2t/D ≈ 1.2–1.6 may turn out to be far from the optimal value for the amplitudes of the AC magnetic field used. Similarly, magnetosomes with an average diameter D = 40–50 nm were covered with shells of various chemical compositions [41], but of sufficiently small thickness, t = 2–6 nm.

_{p}= 4–6) within a tumor. A constant magnetic field can be temporarily applied during the procedure of introducing magnetosome chains into a tumor to provide correct chain orientation.

## 5. Conclusions

_{0}. It is shown that assemblies of magnetosome chains of different lengths have comparable SAR values, provided that the number of particles in the chain exceeds N

_{p}= 4–5. However, the SAR of an oriented chain assembly significantly decreases for large angles θ > 50° of the magnetic field direction with respect to chain axis. In addition, the SAR value of the oriented assembly of magnetosome chains also decreases rapidly with an increasing average density of the assembly due to increase in the intensity of a magneto-dipole interaction between the nanoparticles belonging to various chains.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Dependence of assembly SAR on the reduced distance a/D between the particle centers for chains with various particle diameters. (

**b**) Evolution of the shape of the low-frequency hysteresis loop for chains of particles with diameter D = 30 nm for various reduced distances: (1) a/D = 1.7, (2) a/D = 1.85, (3) a/D = 2.4, (4) a/D = 3.0.

**Figure 2.**(

**a**) Frequency dependence of SAR for assembly of magnetosome chains with particle diameter D = 30 nm in an AC magnetic field with an amplitude H

_{0}= 50 Oe. (

**b**) Dependence of SAR on the distance a between the particle centers for particles with diameter D = 20 nm for various amplitudes of AC magnetic field: (1) H

_{0}= 50 Oe, (2) H

_{0}= 100 Oe, (3) H

_{0}= 150 Oe.

**Figure 3.**(

**a**) Distribution of the components of the interaction field depending on the position of the nanoparticle in the chain. (

**b**) Dependence of the SAR of a dilute chain assembly on the number of particles N

_{p}in the chainswith particles of different diameters, D = 20–40 nm, assuming the optimal ratios a/D for various chains.

**Figure 4.**(

**a**) Dependence of the assembly SAR on the angle of AC magnetic field with respect to the chain axis for assemblies of particles of different diameters. (

**b**) The evolution of low frequency hysteresis loops for assembly of particles with diameter D = 30 nm for different directions of applied magnetic field with respect to the common chain axis: (1) θ = 0, (2) θ = 30°, (3) θ = 40°, (4) θ = 60°.

**Figure 5.**(

**a**) Low-frequency hysteresis loops of dilute assemblies of cylindrical clusters of interacting chains of nanoparticles depending on the number of chains located inside the cluster: (1) N

_{ch}= 4, (2) N

_{ch}= 6, (3) N

_{ch}= 8, (4) N

_{ch}= 10. (

**b**) The model of random oriented cluster of magnetosome chains used in the calculations.

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**MDPI and ACS Style**

Usov, N.A.; Gubanova, E.M.
Application of Magnetosomes in Magnetic Hyperthermia. *Nanomaterials* **2020**, *10*, 1320.
https://doi.org/10.3390/nano10071320

**AMA Style**

Usov NA, Gubanova EM.
Application of Magnetosomes in Magnetic Hyperthermia. *Nanomaterials*. 2020; 10(7):1320.
https://doi.org/10.3390/nano10071320

**Chicago/Turabian Style**

Usov, Nikolai A., and Elizaveta M. Gubanova.
2020. "Application of Magnetosomes in Magnetic Hyperthermia" *Nanomaterials* 10, no. 7: 1320.
https://doi.org/10.3390/nano10071320