**1. Introduction**

The Magnetic Hyperthermia is one of the most promising therapies in the cancer treatment [1]. The malignant tissues are destroyed when their temperature reach a therapeutic

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

hyperthermic temperature in the range 40–46°C [2, 3]. The main problem of this technique is to understand and to control as precisely as possible the temperature field developed by the magnetic systems injected within malignant tissues when the external alternating magnetic fields are applied.

a ferrofluid volume *Vf*

(μl/min) using a needle of a syringe with the radius *ro*

defined by the following distribution function [7]:

**2.1. The radial distribution of the MNP concentration**

the continuity equation in the spherical coordinates:

The radial velocity of the particles—the component of the velocity vector: *v*

**Figure 1.** The geometric configuration of a malignant and healthy tissue structure.

∇ ∙*v*

*vr* <sup>=</sup> \_\_*<sup>B</sup>*

*<sup>g</sup>*[R] <sup>=</sup> \_\_\_\_\_ <sup>1</sup>

R is the particle radius, ln[*R*<sup>0</sup>

etry starts from the center O where is localized the injection site (IS).

*R* √ \_\_\_ <sup>2</sup>*<sup>π</sup>* exp[<sup>−</sup>

*Qv*

was injected with the volumetric flow rate (ferrofluid infusion rate)

∝

*g*[R] *dR* = 1

Modeling of the Temperature Field in the Magnetic Hyperthermia

<sup>→</sup> = 0 (1)

*<sup>r</sup>* <sup>2</sup> (1.1)

<sup>→</sup>(*vr*

, <sup>0</sup>, <sup>0</sup>) is given by:

In this analysis, MNP with different sizes were considered. A lognormal distribution was

[ln(*<sup>R</sup>* ⁄ *<sup>R</sup>*0)]2 \_\_\_\_\_\_\_

The ferrofluid (composed by small magnetite particles and water) was considered an incompressible diluted colloidal fluid with the small concentration (*c* ≤ 5% by volume). The presence of the small magnetite particles does not significantly affect the transport properties of the fluid [8]. The velocity of the magnetic particles within tissues was computed as a solution of

<sup>2</sup> *<sup>σ</sup>*<sup>2</sup> ] and <sup>∫</sup><sup>0</sup>

] is the median and *σ* is the standard deviation of R.

. The ferrofluid flow within this geom-

http://dx.doi.org/10.5772/intechopen.71364

307

Some experimental data realized in a tissue equivalent (agarose gel) evidences the particle diffusion within the tissue after their injection [4, 5]. The diffusion-convection and deposition of the particles have a strong influence on the radial particle distribution within the tissue volume and as a consequence on the temperature field in tissues [6].

In this paper, the temperature within a malignant tissue surrounded by a healthy tissue was studied considering the radial magnetic nanoparticles (MNP) concentration as an effect of the ferrofluid injection at the center of tumor. The MNP with different sizes having a lognormal particle size distribution were considered. The temperature developed by the magnetic systems in the external time-dependent magnetic field was analyzed for different values of the parameters. During the injection process of the particles within the tissues, their convection and deposition influences strongly the concentration of the particles. An analytical model was developed to predict the temperature field for different important parameters as: (i) ferrofluid infusion rates, (ii) particle zeta potential and (iii) optimal particle dosages. The results were compared with a numerical model in Comsol Multiphysics and Matlab. The values of temperatures computed using the analytical and numerical models in the same conditions were in very good agreement.
