**2.1. Mechanisms of heat dissipation of MNPs**

To turn MNPs into heaters, they are subjected to an oscillating electromagnetic field, where the field's direction changes cyclically. There are various theories which explain the reasons for the heating of the MNPs when subjected to an oscillating electromagnetic field [Brusentsova *et al.* (2005), Jo´zefczak and Skumiel (2007), Kim *et al.* (2008), Golneshan and Lahonian (2010), Golneshan and Lahonian (2011c)].

There exist at least four different mechanisms by which magnetic materials can generate heat in an alternating field [Nedelcu (2008)]:


Relaxation losses in single-domain MNPs fall into two modes: rotational (Brownian) mode and Néel mode. The principle of heat generation due to each individual mode is shown in Figure 2.

In the Néel mode, the magnetic moment originally locked along the crystal easy axis rotates away from that axis towards the external field. The Néel mechanism is analogous to the hysteresis loss in multi-domain MNPs whereby there is an 'internal friction' due to the

**Figure 1.** Schematic of magnetic fluid hyperthermia process.

Two techniques are currently used to deliver the MNPs to the tumor. The first is to deliver particles to the tumor vasculature [Matsuki and Yanada (1994)] through its supplying artery; however, this method is not effective for poorly perfused tumors. Furthermore, for a tumor with an irregular shape, inadequate MNPs distribution may cause under-dosage of heating in the tumor or overheating of the normal tissue. The second approach, is to directly inject MNPs into the extracellular space in the tumors. The MNPs diffuse inside the tissue after injection of nanofluid. If the tumor has an irregular shape, multi-site injection can be

The nanofluid injection volume as well as infusion flow rate of nanofluid are important factors in dispersion and concentration of the MNPs, within the tissue. A successful MFH treatment is substantially dependent on the MNPs distribution in the tissue [Bagaria and Johnson (2005), Salloum *et al.* (2008a), Salloum *et al.* (2008b), Lin and Liu (2009), Bellizzi and

In MFH, after introducing the MNPs into the tumor (Figure 1), an alternating magnetic field is applied. This causes an increase in the tumor temperature and subsequent tumor regression. The temperature that can be achieved in the tissue strongly depends on the properties of the magnetic material used, the frequency and the strength of the applied magnetic field, duration of application of the magnetic field, and dispersion of the MNPs

To turn MNPs into heaters, they are subjected to an oscillating electromagnetic field, where the field's direction changes cyclically. There are various theories which explain the reasons for the heating of the MNPs when subjected to an oscillating electromagnetic field [Brusentsova *et al.* (2005), Jo´zefczak and Skumiel (2007), Kim *et al.* (2008), Golneshan and

There exist at least four different mechanisms by which magnetic materials can generate

Relaxation losses in single-domain MNPs fall into two modes: rotational (Brownian) mode and Néel mode. The principle of heat generation due to each individual mode is shown in Figure 2. In the Néel mode, the magnetic moment originally locked along the crystal easy axis rotates away from that axis towards the external field. The Néel mechanism is analogous to the hysteresis loss in multi-domain MNPs whereby there is an 'internal friction' due to the

exploited to cover the entire target region [Salloum *et al.* (2008a)].

Bucci (2010), Golneshan and Lahonian (2011a)].

**2.1. Mechanisms of heat dissipation of MNPs** 

Lahonian (2010), Golneshan and Lahonian (2011c)].

1. Generation of eddy currents in magnetic particles with size >1μ, 2. Hysteresis losses in magnetic particles >1μ and multidomain MNPs, 3. Relaxation losses in 'superparamagnetic' single-domain MNPs,

heat in an alternating field [Nedelcu (2008)]:

4. Frictional losses in viscous suspensions.

**2. Heat dissipation of MNPs** 

within the tissue.

movement of the magnetic moment in an external field that results in heat generation. In the Brownian mode, the whole particle oscillates towards the field with the moment locked along the crystal axis under the effect of a thermal force against a viscous drag in a suspending medium. This mechanism essentially represents the mechanical friction component in a given suspending medium [Nedelcu (2008)].

**Figure 2.** Relaxation mechanisms of MNPs in Magnetic Fluid. a) Brownian relaxation, entire particle rotates in fluid; b) Néel relaxation, direction of magnetization rotates in core. The structure of MNP: core (inner), shell (outer). The arrow inside the core represents the direction of magnetization.

Power dissipation of MNPs in an alternating magnetic field is expressed as [Rosensweig (2002), Nedelcu (2008)]:

$$P = \pi \mu\_0 \chi\_0 H\_0^2 f \frac{2\pi f \tau}{1 + (2\pi f \tau)^2} \tag{1}$$

$$
\tau^{-1} = \tau\_N^{-1} + \tau\_B^{-1} \tag{2}
$$

$$
\pi\_N = \frac{\sqrt{\pi}}{2} \tau\_0 \frac{\exp(\Gamma)}{\sqrt{\Gamma}} \tag{3}
$$

$$
\pi\_B = \frac{3\eta V\_H}{kT} \tag{4}
$$

$$V\_M = \frac{\pi D^3}{6} \tag{5}$$

$$V\_H = \frac{\pi (D + 2\delta)^3}{6} \tag{6}$$

$$\mathbf{x}\_0 = \mathbf{x}\_l \frac{3}{\xi} \Big(\coth \xi - \frac{1}{\xi}\Big) \tag{7}$$

$$\alpha\_l = \frac{\mu\_0 \phi M\_d^2 V\_M}{3kT} \tag{8}$$


Diffusion of Magnetic Nanoparticles Within a Biological Tissue During Magnetic Fluid Hyperthermia 135

*B0=50 mT* 

*φ=2.0e-5 δ=1 nm*

> *○ f=300 kHz □ f=200 kHz ∆ f=100 kHz*

*○ δ=0.0 nm ◊ δ=1.0 nm × δ=2.0 nm*

*µ=1.0e-3 kg/(m.s)* 

*B0=50 mT f=300 kHz* 

*φ=2.0e-5*

*µ=1.0e-3 kg/(m.s)* 

Figures 4 to 7 show that dispersion and concentration of MNPs inside the tissue are important factors in heat dissipation of MNPs and temperature distribution inside the tumor and its surrounding healthy tissue. Also, the effect of concentration of MNPs is comparable with the effects of induction and frequency of the magnetic field on the maximum power dissipation. Therefore, study of the MNPs diffusion and concentration,

**Figure 5.** Dependence of power dissipation on ��[Lahonian and Golneshan (2011)].

0 10 20 30 40

**D (nm)**

0 10 20 30 40

**D (nm)**

**Figure 6.** Dependence of power dissipation on ��[Lahonian and Golneshan (2011)].

possesses a high degree of importance.

0

0

1

2

**P (10**⁵

**W/m³)**

3

4

5

1

2

**P (10**⁵

**W/m³)**

3

4

**Figure 3.** Power dissipations as a function of particle diameter for various MNPs [Lahonian and Golneshan (2011)].

**Figure 4.** Dependence of power dissipation on �� [Lahonian and Golneshan (2011)].

Figures 4 to 7 show that dispersion and concentration of MNPs inside the tissue are important factors in heat dissipation of MNPs and temperature distribution inside the tumor and its surrounding healthy tissue. Also, the effect of concentration of MNPs is comparable with the effects of induction and frequency of the magnetic field on the maximum power dissipation. Therefore, study of the MNPs diffusion and concentration, possesses a high degree of importance.

134 Hyperthermia

Golneshan (2011)].

0

1

2

**P (10**⁵

**W/m³)**

3

4

5

**Figure 3.** Power dissipations as a function of particle diameter for various MNPs [Lahonian and

0 10 20 30 40

**D (nm)**

**Figure 4.** Dependence of power dissipation on �� [Lahonian and Golneshan (2011)].

0 10 20 30 40

*○ B0=80 mT □ B0=50 mT ∆ B0=30 mT*

*µ=1.0e-3 kg/(m.s)*

○ FCC FePt MNPs □ Magnetite MNPs ∆ Maghemite MNPs

*µ=1.0e-3 kg/(m.s)* 

*B0=50 mT f=300 kHz* 

*φ=2.0e-5 δ=1 nm* 

*f=300 kHz*

*φ=2.0e-5 δ=1 nm*

**D (nm)**

0

2

**P (10**⁵

**W/m³)**

4

6

**Figure 5.** Dependence of power dissipation on ��[Lahonian and Golneshan (2011)].

**Figure 6.** Dependence of power dissipation on ��[Lahonian and Golneshan (2011)].

**Figure 7.** Dependence of power dissipation on ��[Lahonian and Golneshan (2011)].
