**3. Diffusion of MNPs within the biological tissue**

The relationship among the MNPs distribution, the blood perfusion, the infusion flow rate, the injection volume of nanofluid, and the tissue structure are not well understood. It is difficult to devise a treatment protocol that enables the optimum distribution of temperature elevation in the tumor. Hence, it is important to quantify the MNPs distribution and heating pattern following the injection regarding the above mentioned factors [Salloum *et al.* (2008b)].

Diffusion in isotropic tissues, can be modeled as [Nicholson (2001)]:

$$\frac{\partial \mathcal{C}}{\partial t} = \mathbf{D}^\* \nabla^2 \mathcal{C} + \mathcal{S}/\varepsilon \tag{9}$$

Diffusion of Magnetic Nanoparticles Within a Biological Tissue During Magnetic Fluid Hyperthermia 137

vicinity of the injection site. Also the particle deposition was greatly affected by the injection rate and amount. Furthermore, the shape of the distribution tended to be more irregular

Due to difficulties in experimental studies, to understand the actual spatial distribution of the MNPs after being injected into the tumor, some numerical simulations have been down.

Diffusion of MNPs inside the tissue was simulated by Golneshan and Lahonian (2011a). A square region with side of 2��� was chosen as the domain of the analysis (Figure 8). Waterbased ferrofluid with a concentration of 3.3% by volume and a particle size of 10��� magnetite MNPs was used in their work. Based on the density of magnetite �52�0 �� �� ⁄ ) and the given ferrofluid concentration, each 0.1��� of ferrofluid contains 1�.3�� of solid iron oxide [Golneshan and Lahonian (2010)]. The ferrofluid infusion flow rates were chosen equal to �� = 10, 20 and 30������� and ferrofluid injection volumes were chosen equal to � = 0.1, 0.2 and 0.3���. Porosity and effective diffusivity were chosen to be equal to � = 0.1 and �<sup>∗</sup> = 2.5 × 10��� ���⁄� respectively [Nicholson (2001), Golneshan and Lahonian (2010)].

Figure 9 shows the concentration of ferrofluid in the tissue for � = 0.2��� and �� = 20�������, for different time intervals after the end of ferrofluid injection. Results show that the concentration of ferrofluid is maximum at the injection site, and decreases rapidly with increasing distance from it. Also, concentration of ferrofluid decreases at the injection area with time and increases in the surrounding of injection site [Golneshan and Lahonian

2 ��

Figure 10 shows volume fraction of MNPs in the tissue for different ferrofluid injection volumes, �� = 20� �� ��� ⁄ , at 20 minutes after the end of ferrofluid injection [Golneshan and

Figure 11 shows the concentration of ferrofluid in the tissue for � = 0.2���, and different infusion flow rates, just 20 minutes after the end of ferrofluid injection. Results show that

with higher infusion flow rate.

(2011a)].

Lahonian (2011a)].

**Figure 8.** Simulation domain of tissue and injection site.

2����

where �, �∗, �, � and � are the volume average concentration of the species, effective diffusivity, mass source density, porosity of the tissue and time, respectively. The effective diffusivity, however, is related to the tortuosity of the tissue, �, and the diffusivity in the absence of the porous medium, � through the following relation:

$$D^\* = D / \lambda^2 \tag{10}$$

Therefore an increase in the tortuosity and a decrease in the porosity have significant effects on reducing the effective mass diffusivity.

Experimental study of Salloum *et al.* (2008a) in a tissue-equivalent agarose gel, showed that the particle concentration was not uniform after the injection and were confined in the vicinity of the injection site. Also the particle deposition was greatly affected by the injection rate and amount. Furthermore, the shape of the distribution tended to be more irregular with higher infusion flow rate.

136 Hyperthermia

(2008b)].

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

The relationship among the MNPs distribution, the blood perfusion, the infusion flow rate, the injection volume of nanofluid, and the tissue structure are not well understood. It is difficult to devise a treatment protocol that enables the optimum distribution of temperature elevation in the tumor. Hence, it is important to quantify the MNPs distribution and heating pattern following the injection regarding the above mentioned factors [Salloum *et al.*

0 10 20 30 40

○ ϕ=4.0e-5 ◊ ϕ=3.0e-5 × ϕ=2.0e-5

*µ=1.0e-3 kg/(m.s)* 

*B0=50 mT f=300 kHz* 

*δ=1 nm*

**D (nm)**

where �, �∗, �, � and � are the volume average concentration of the species, effective diffusivity, mass source density, porosity of the tissue and time, respectively. The effective diffusivity, however, is related to the tortuosity of the tissue, �, and the diffusivity in the

Therefore an increase in the tortuosity and a decrease in the porosity have significant effects

Experimental study of Salloum *et al.* (2008a) in a tissue-equivalent agarose gel, showed that the particle concentration was not uniform after the injection and were confined in the

�� = D∗∇���� �⁄ (9)

�<sup>∗</sup> =� �� ⁄ (10)

**3. Diffusion of MNPs within the biological tissue** 

0

2

4

**P (10⁵**

**W/m³)**

6

8

10

Diffusion in isotropic tissues, can be modeled as [Nicholson (2001)]:

absence of the porous medium, � through the following relation:

on reducing the effective mass diffusivity.

��

Due to difficulties in experimental studies, to understand the actual spatial distribution of the MNPs after being injected into the tumor, some numerical simulations have been down.

Diffusion of MNPs inside the tissue was simulated by Golneshan and Lahonian (2011a). A square region with side of 2��� was chosen as the domain of the analysis (Figure 8). Waterbased ferrofluid with a concentration of 3.3% by volume and a particle size of 10��� magnetite MNPs was used in their work. Based on the density of magnetite �52�0 �� �� ⁄ ) and the given ferrofluid concentration, each 0.1��� of ferrofluid contains 1�.3�� of solid iron oxide [Golneshan and Lahonian (2010)]. The ferrofluid infusion flow rates were chosen equal to �� = 10, 20 and 30������� and ferrofluid injection volumes were chosen equal to � = 0.1, 0.2 and 0.3���. Porosity and effective diffusivity were chosen to be equal to � = 0.1 and �<sup>∗</sup> = 2.5 × 10��� ���⁄� respectively [Nicholson (2001), Golneshan and Lahonian (2010)].

Figure 9 shows the concentration of ferrofluid in the tissue for � = 0.2��� and �� = 20�������, for different time intervals after the end of ferrofluid injection. Results show that the concentration of ferrofluid is maximum at the injection site, and decreases rapidly with increasing distance from it. Also, concentration of ferrofluid decreases at the injection area with time and increases in the surrounding of injection site [Golneshan and Lahonian (2011a)].

**Figure 8.** Simulation domain of tissue and injection site.

Figure 10 shows volume fraction of MNPs in the tissue for different ferrofluid injection volumes, �� = 20� �� ��� ⁄ , at 20 minutes after the end of ferrofluid injection [Golneshan and Lahonian (2011a)].

Figure 11 shows the concentration of ferrofluid in the tissue for � = 0.2���, and different infusion flow rates, just 20 minutes after the end of ferrofluid injection. Results show that the increasing infusion flow rate, increases concentration of ferrofluid in the vicinity of the injection site while decreasing the concentration in the layers far from the injection site [Golneshan and Lahonian (2011a)].

Diffusion of Magnetic Nanoparticles Within a Biological Tissue During Magnetic Fluid Hyperthermia 139

**Figure 11.** Concentration of ferrofluid in the tissue for � � 0�2���, and different infusion flow rates, just

0

0.0115 0.012 0.0125 0.013 0.0135

*□ 10 µlit/min ∆ 20 µlit/min ○ 30 µlit/min*

50

100

Figure 12 shows the concentration of ferrofluid in the tissue for � � 0�2���, and different infusion flow rates, just 20 minutes after the end of ferrofluid injection. Results show that the increasing infusion flow rate, increases concentration of ferrofluid in the vicinity of the injection site but decreases the concentration in the layers far from the injection site

**4. Diffusion of MNPs in a biological tissue for mono and multi-site** 

Golneshan and Lahonian (2011a) studied diffusion of MNPs in a biological tissue for irregular tumors. A 2 � 2��� tissue with an irregular tumor inside, was chosen as the

They considered multi-site injection as shown in Figure 13d and divided the irregular tumor almost into four equal sections. In each injection site, one fourth the amount of 0�2��� ferrofluid was injected. Figure 14 shows the concentration of ferrofluid for infusion flow rate of �� � 20� �� ��� ⁄ , at the end of ferrofluid injection [Golneshan and Lahonian

20 minutes after the end of ferrofluid injection [Golneshan and Lahonian (2011a)].

0.01 0.011 0.012 0.013 0.014 0.015

**x (m)**

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

**injection for irregular tumors** 

domain of the analysis (Figures 13a).

(2011a)].

0

200

400

**C (kg/m3)**

**Figure 9.** Concentration of ferrofluid in the tissue, for different time intervals after the end of ferrofluid injection (� � ������ and �� � ���������) [Golneshan and Lahonian (2011a)].

**Figure 10.** Ferrofluid concentration for �� � ��� �� ��� ⁄ , and different ferrofluid injection volumes, just 20 minutes after the end of ferrofluid injection [Golneshan and Lahonian (2011a)].

[Golneshan and Lahonian (2011a)].

the increasing infusion flow rate, increases concentration of ferrofluid in the vicinity of the injection site while decreasing the concentration in the layers far from the injection site

> □ 10 minutes ◊ 20 minutes ∆ 30 minutes × 40 minutes + 50 minutes ○ 60 minutes

> > *□ V=0.1 cc ∆ V=0.2 cc ○ V=0.3 cc*

**Figure 9.** Concentration of ferrofluid in the tissue, for different time intervals after the end of ferrofluid

**x (m)**

0.01 0.012 0.014

**Figure 10.** Ferrofluid concentration for �� � ��� �� ��� ⁄ , and different ferrofluid injection volumes, just

**x (m)**

0.01 0.012 0.014

20 minutes after the end of ferrofluid injection [Golneshan and Lahonian (2011a)].

injection (� � ������ and �� � ���������) [Golneshan and Lahonian (2011a)].

0

0

200

400

**C (kg/m3)**

600

800

200

400

**C (kg/m3)**

600

800

**Figure 11.** Concentration of ferrofluid in the tissue for � � 0�2���, and different infusion flow rates, just 20 minutes after the end of ferrofluid injection [Golneshan and Lahonian (2011a)].

Figure 12 shows the concentration of ferrofluid in the tissue for � � 0�2���, and different infusion flow rates, just 20 minutes after the end of ferrofluid injection. Results show that the increasing infusion flow rate, increases concentration of ferrofluid in the vicinity of the injection site but decreases the concentration in the layers far from the injection site [Golneshan and Lahonian (2010), Golneshan and Lahonian (2011a)].
