**2. The analytical temperature model**

Modeling the heat transport within the malignant tissues is an important element in the Magnetic Hyperthermia. The development of an analytical model for a complex system or process is usually an important breakthrough with a strong impact in the field.

In this case we are presenting an analytical model which will be a powerful analysis tool for hyperthermia treatment planning due to its ability to predict accurately the temperature field within the malignant tissues. The model will provide a tool to study the main parameters which influence significantly the temperature field and to give a tool to optimize them. For each patient, an individual therapy planning is required in correlation with the tumor location, geometry, shape and size. The elaboration of a patient model by segmentation of images from computerized tomography or magnetic resonance imaging scans is the first step—and the most important one—of hyperthermia treatment planning. This segmentation is used to compute the temperature field in the tumor located in a patient organ.

In our simulations a spherical concentric configuration composed of a malignant tissue surrounded by a healthy tissue was considered (**Figure 1**). The malignant tissue has a radius *R*<sup>1</sup> and the healthy region has the shell thickness *R*<sup>2</sup>  − *R*<sup>1</sup> . At the center O of this geometric structure, a ferrofluid volume *Vf* was injected with the volumetric flow rate (ferrofluid infusion rate) *Qv* (μl/min) using a needle of a syringe with the radius *ro* . The ferrofluid flow within this geometry starts from the center O where is localized the injection site (IS).

In this analysis, MNP with different sizes were considered. A lognormal distribution was defined by the following distribution function [7]:

$$g[\text{R}] = \frac{1}{\sigma \text{R} \sqrt{2\pi}} \exp\left[-\frac{[\ln(\text{R} \,\text{Å})]^2}{2\sigma^2}\right] \text{ and } \int\_0^\kappa g[\text{R}] \,d\text{R} = 1$$

R is the particle radius, ln[*R*<sup>0</sup> ] is the median and *σ* is the standard deviation of R.

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

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

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

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

Modeling the heat transport within the malignant tissues is an important element in the Magnetic Hyperthermia. The development of an analytical model for a complex system or

In this case we are presenting an analytical model which will be a powerful analysis tool for hyperthermia treatment planning due to its ability to predict accurately the temperature field within the malignant tissues. The model will provide a tool to study the main parameters which influence significantly the temperature field and to give a tool to optimize them. For each patient, an individual therapy planning is required in correlation with the tumor location, geometry, shape and size. The elaboration of a patient model by segmentation of images from computerized tomography or magnetic resonance imaging scans is the first step—and the most important one—of hyperthermia treatment planning. This segmentation is used to compute the temperature field in the tumor located in a patient organ.

In our simulations a spherical concentric configuration composed of a malignant tissue surrounded by a healthy tissue was considered (**Figure 1**). The malignant tissue has a radius *R*<sup>1</sup>

 − *R*<sup>1</sup>

. At the center O of this geometric structure,

process is usually an important breakthrough with a strong impact in the field.

volume and as a consequence on the temperature field in tissues [6].

magnetic fields are applied.

306 Numerical Simulations in Engineering and Science

very good agreement.

**2. The analytical temperature model**

and the healthy region has the shell thickness *R*<sup>2</sup>

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 the continuity equation in the spherical coordinates:

$$\nabla \cdot \vec{v} = 0 \tag{1}$$

The radial velocity of the particles—the component of the velocity vector: *v* <sup>→</sup>(*vr* , <sup>0</sup>, <sup>0</sup>) is given by:

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

where the variable r defines the radial distance from the IS localized at the center of this geometry. The constant *<sup>B</sup>* <sup>=</sup> *<sup>Q</sup>*\_\_\_*<sup>v</sup> <sup>π</sup>* was computed considering the ferrofluid velocity at the tip of the needle as:

$$
\mathcal{U} = \frac{\mathcal{Q}\_r}{\mathcal{S}\_{\text{rooted}}} = \frac{B}{r\_0^2} \tag{1.2}
$$

The expression ∇ ∙(*u*

*f i Ci*

fusion and *k*

meability *Ki*

*C<sup>i</sup>*

(i) *C*<sup>2</sup>

expression C1

→*i Ci*

rate of the particles on the solid phase *k*

*kf*

diameter *D*, the radial velocity *vr*

Ai \_\_\_\_\_\_\_\_\_\_ 2r √ \_

*mi* = −

<sup>r</sup> ){(**const1**)

*i*

**Boundaries conditions:** The constants, (**const1**)

(*r*) <sup>=</sup> ( <sup>e</sup><sup>−</sup> \_\_

where the expressions *mi*

The *collector efficiency <sup>η</sup><sup>s</sup>*

(A1.2) from **Appendix 1**.

lowing boundary conditions:

*D*<sup>1</sup>

 = Cmax.

mainly on: (i) the ferrofluid infusion rate Qv

) and particle zeta potential *ζ<sup>p</sup>*

the following solutions (**Appendix 1** - relations (A1.15)):

and *Ai*

 = 0 on the external boundary of the geometry (r = R<sup>2</sup>

(ii) Neumann boundary condition at the all inner interfaces:

C1(r = R1) = C2(r = R1)

(iii) at the injection site (IS), at the top of the needle (r = r<sup>o</sup>

<sup>i</sup> Bessel I[√

are:

3(1 − *ε<sup>i</sup>* ) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 2 *ε<sup>i</sup> dc ηs <sup>i</sup>* \_\_\_\_ *Qv π Di*

<sup>∗</sup> \_\_\_ ∂ C1 ∂r |r=R1

) describes the particle convection, the term <sup>∇</sup> <sup>∙</sup> (*Di*

The deposition processes play an important role in the spatial distribution of the particles. The temperature field within geometry is strongly dependent on the volumetric deposition

were computed considering the superposition of the effects developed by the hydrodynamic forces, van der Waals interactions, gravity effect and the repulsive electrostatic double layer forces. The particle deposition on the cellular structure of tissues depends on the particle

> \_\_\_\_\_\_ 1 − 4 mi

, \_\_ Ai

i

= *D*<sup>2</sup> <sup>∗</sup> \_\_\_ ∂ C2 ∂r |r=R1

The radial distribution of the MNP concentrations computed as solutions of Eq. (5) depends

ity, permeability and particle size (**Table 1**). The convection, diffusion and deposition of the

2r ] <sup>+</sup> (**const2**)

<sup>∗</sup> and *Ai* <sup>=</sup> \_\_\_\_ *Qv*

and (**const**2)<sup>i</sup>

*π Di* ∗.

as result of the electrostatic repulsive forces is given by the relation

);

*f i Ci*

*<sup>i</sup>* <sup>=</sup> 3(1 <sup>−</sup> *<sup>ε</sup><sup>i</sup>* ) \_\_\_\_\_\_\_\_\_\_ 2 *ε<sup>i</sup> dc ηs i vr*


. The coefficients *k*

, (i = 1, 2)

*f i*

of the particles, tissues characteristics (porosity *ε<sup>i</sup>* and per-

(**Appendix 1**) [10–16]. At equilibrium, Eq. (5) have

<sup>i</sup> Bessel K[√

\_\_\_\_\_\_ <sup>1</sup> <sup>−</sup> <sup>4</sup> mi , \_\_

, were computed using the fol-

) the concentration has the particular

; (ii) tissue and particle characteristics as poros-

Ai 2r ]} (6)

<sup>∗</sup> ∇ *Ci*

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

Modeling of the Temperature Field in the Magnetic Hyperthermia

) - the particle dif-

309

*Sneedle* = *π r* 0 2 is the needle cross sectional area. The radial velocity of the particles depends on the volumetric flow rate *Qv* (μl/min) and the radial variable r:

$$
\upsilon\_r = \frac{Q\_\upsilon}{\pi r^2} \tag{2}
$$

Local velocity of the particles (2) depends on the pressure gradient developed within geometry as result of the ferrofluid injection process. The ferrofluid flow through tissues was modeled using the Darcy's equation [8]:

$$
\nabla P\_i = -\frac{\varepsilon\_i \mu}{K\_i} \vec{\upsilon} \text{ ( $i = 1, 2$ )}\tag{3}
$$

In this analysis, the index *i =* 1 defines the tumor (malignant tissue) and the index *i =* 2 defines the healthy tissue. The expression of the pressure in the malignant tissue is *P*<sup>1</sup> (*r*):

$$P\_1(r) = \frac{\varepsilon\_1 \mu}{K\_1} \left[ \frac{1}{r} - \frac{1}{R\_1} \right] \frac{\mathcal{Q}\_v}{\pi t} + \frac{\varepsilon\_2 \mu}{K\_2} \left[ \frac{1}{R\_1} - \frac{1}{R\_2} \right] \frac{\mathcal{Q}\_v}{\pi t}, \quad r\_o \le r \le R\_1. \tag{4.1}$$

and *P*<sup>2</sup> (*r*) in the healthy region:

$$P\_2(r) = \frac{\varepsilon\_z \mu}{K\_2} \left[ \frac{1}{7} - \frac{1}{R\_2} \right] \frac{Q\_v}{7^\circ} \quad R\_1 \le r \le R\_2 \tag{4.2}$$

was computed solving the Darcy's equation (3) for the concentric tissues. On the external border of the geometry, the pressure is zero, *P*<sup>2</sup> (*r* = *R*<sup>2</sup> ) = 0. The pressure developed in this geometric configuration depends significantly on the parameter *Qv* , ferrofluid viscosity *μ* and tissues characteristics (porosity *ε<sup>i</sup>* and permeability *Ki* ).

The mass concentrations of the particles, *Ci*  = *Ci* (*r*) (*i* = 1, 2) (expressed in mg/cm<sup>3</sup> ) are the solutions of the modified convection-diffusion equation [8, 9]:

$$\frac{\partial \mathbf{C}\_{\rangle}}{\partial t} + \nabla \cdot \{\vec{\mu}\_{i} \, \mathbf{C}\_{i}\} = \nabla \cdot \{\mathbf{D}\_{i}^{\*} \nabla \mathbf{C}\_{i}\} - k\_{j}^{i} \mathbf{C}\_{i} \tag{5}$$

where *k f i* represent the values of the deposition rate coefficients of the particles within the malignant and healthy tissues which were computed using the relations (A1.3)–(A1.5) from **Appendix 1**.

The expression ∇ ∙(*u* →*i Ci* ) describes the particle convection, the term <sup>∇</sup> <sup>∙</sup> (*Di* <sup>∗</sup> ∇ *Ci* ) - the particle diffusion and *k f i Ci* - the mean volumetric deposition rate of the particle on the solid phase.

The deposition processes play an important role in the spatial distribution of the particles. The temperature field within geometry is strongly dependent on the volumetric deposition rate of the particles on the solid phase *k f i Ci* . The coefficients *k f i*

$$k\_{\!\!\!/}^{\!\!\!/} = \frac{3(1-\varepsilon\_{\!\!\!/})}{2\ \varepsilon\_{\!\!\!/}d\_{\!\!\!/}}\eta\_{\!\!\!/}v\_{\!\!\!/} \begin{pmatrix} \text{i } = \ 1,2 \end{pmatrix}$$

were computed considering the superposition of the effects developed by the hydrodynamic forces, van der Waals interactions, gravity effect and the repulsive electrostatic double layer forces. The particle deposition on the cellular structure of tissues depends on the particle diameter *D*, the radial velocity *vr* of the particles, tissues characteristics (porosity *ε<sup>i</sup>* and permeability *Ki* ) and particle zeta potential *ζ<sup>p</sup>* (**Appendix 1**) [10–16]. At equilibrium, Eq. (5) have the following solutions (**Appendix 1** - relations (A1.15)):

$$C\_{\circ}(r) = \left(\frac{e^{-\frac{\lambda}{2\pi}}}{\sqrt{\pi}}\right) \left[ \mathbf{(const1)}\_{\circ}, \text{ Bessel } \mathbf{I}\left[\sqrt{1-4\,\mathbf{m}\_{\circ}}, \frac{\mathbf{A}\_{\circ}}{2\pi}\right] + \mathbf{(const2)}\_{\circ}, \text{ Bessel } \mathbf{K}\left[\sqrt{1-4\,\mathbf{m}\_{\circ}}, \frac{\mathbf{A}\_{\circ}}{2\pi}\right] \right] \tag{6}$$

where the expressions *mi* and *Ai* are:

where the variable r defines the radial distance from the IS localized at the center of this

*S*needle

(μl/min) and the radial variable r:

= \_\_*<sup>B</sup> r*0

is the needle cross sectional area. The radial velocity of the particles depends on the

*Qv*

Local velocity of the particles (2) depends on the pressure gradient developed within geometry as result of the ferrofluid injection process. The ferrofluid flow through tissues was mod-

> *ε<sup>i</sup> μ* \_\_\_\_\_\_\_\_ *Ki v*

In this analysis, the index *i =* 1 defines the tumor (malignant tissue) and the index *i =* 2 defines

was computed solving the Darcy's equation (3) for the concentric tissues. On the external

 = *Ci*

) = ∇ ∙(*Di*

malignant and healthy tissues which were computed using the relations (A1.3)–(A1.5) from

(*r* = *R*<sup>2</sup>

).

∗ ∇*Ci* ) − *kf*

represent the values of the deposition rate coefficients of the particles within the

*Qv<sup>π</sup>* <sup>+</sup> *<sup>ε</sup>*<sup>2</sup> *<sup>μ</sup>* \_\_\_ *<sup>K</sup>*<sup>2</sup> [ \_\_1 *R*1 − \_\_1 *<sup>R</sup>*<sup>2</sup> ] \_\_\_

the healthy tissue. The expression of the pressure in the malignant tissue is *P*<sup>1</sup>

*<sup>π</sup>* was computed considering the ferrofluid velocity at the tip of

<sup>2</sup> (1.2)

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

<sup>→</sup> (*i* = 1, 2) (3)

*<sup>π</sup> R*<sup>1</sup> ≤ *r* ≤ *R*<sup>2</sup> (4.2)

) = 0. The pressure developed in this

, ferrofluid viscosity *μ* and

*<sup>i</sup> Ci* (5)

) are the solu-

*Qv<sup>π</sup>* , *ro* <sup>≤</sup> *<sup>r</sup>* <sup>≤</sup> *<sup>R</sup>*<sup>1</sup>

(*r*) (*i* = 1, 2) (expressed in mg/cm<sup>3</sup>

(*r*):

. (4.1)

geometry. The constant *<sup>B</sup>* <sup>=</sup> *<sup>Q</sup>*\_\_\_*<sup>v</sup>*

308 Numerical Simulations in Engineering and Science

volumetric flow rate *Qv*

*P*<sup>1</sup>

and *P*<sup>2</sup>

where *k f i*

**Appendix 1**.

eled using the Darcy's equation [8]:

(*r*) in the healthy region:

*P*<sup>2</sup>

<sup>∂</sup>*C*\_\_\_*<sup>i</sup>*

∇*Pi* = −

(*r*) <sup>=</sup> *<sup>ε</sup>*<sup>1</sup> *<sup>μ</sup>* \_\_\_

border of the geometry, the pressure is zero, *P*<sup>2</sup>

The mass concentrations of the particles, *Ci*

tissues characteristics (porosity *ε<sup>i</sup>* and permeability *Ki*

tions of the modified convection-diffusion equation [8, 9]:

*<sup>K</sup>*<sup>1</sup> [ \_\_1 *<sup>r</sup>* <sup>−</sup> \_\_1 *<sup>R</sup>*<sup>1</sup> ] \_\_\_

> (*r*) <sup>=</sup> *<sup>ε</sup>*<sup>2</sup> *<sup>μ</sup>* \_\_\_ *<sup>K</sup>*<sup>2</sup> [ \_\_1 *<sup>r</sup>* <sup>−</sup> \_\_1 *<sup>R</sup>*<sup>2</sup> ] \_\_\_ *Qv*

geometric configuration depends significantly on the parameter *Qv*

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> <sup>∇</sup> <sup>∙</sup>(*<sup>u</sup>* → *i Ci*

*<sup>U</sup>* <sup>=</sup> \_\_\_\_\_ *Qv*

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

the needle as:

*Sneedle* = *π r* 0 2

$$m\_{\boldsymbol{i}} = -\frac{3(1-\varepsilon\_{\boldsymbol{i}})}{2\operatorname{\varepsilon}\_{\boldsymbol{i}}\operatorname{d}\_{\boldsymbol{c}}}\eta\_{\boldsymbol{s}}^{\boldsymbol{i}}\frac{Q\_{\boldsymbol{v}}}{\pi\operatorname{D}\_{\boldsymbol{i}}^{\*}}\text{ and }A\_{\boldsymbol{i}} = \frac{Q\_{\boldsymbol{v}}}{\pi\operatorname{D}\_{\boldsymbol{i}}^{\*}}.$$

The *collector efficiency <sup>η</sup><sup>s</sup> i* as result of the electrostatic repulsive forces is given by the relation (A1.2) from **Appendix 1**.

**Boundaries conditions:** The constants, (**const1**) i and (**const**2)<sup>i</sup> , were computed using the following boundary conditions:

(i) *C*<sup>2</sup>  = 0 on the external boundary of the geometry (r = R<sup>2</sup> );

(ii) Neumann boundary condition at the all inner interfaces:

$$\mathbf{C\_1(r=R\_1) = C\_2(r=R\_1)}$$

$$\left. D\_{\mathbf{i}}^{\*} \frac{\partial \mathbf{C}\_{\mathbf{i}}}{\partial \mathbf{r}} \right|\_{\mathbf{r} = \mathbf{R}\_{\mathbf{i}}} = \left. D\_{\mathbf{i}}^{\*} \frac{\partial \mathbf{C}\_{\mathbf{i}}}{\partial \mathbf{r}} \right|\_{\mathbf{r} = \mathbf{R}\_{\mathbf{i}}}$$

(iii) at the injection site (IS), at the top of the needle (r = r<sup>o</sup> ) the concentration has the particular expression C1  = Cmax.

The radial distribution of the MNP concentrations computed as solutions of Eq. (5) depends mainly on: (i) the ferrofluid infusion rate Qv ; (ii) tissue and particle characteristics as porosity, permeability and particle size (**Table 1**). The convection, diffusion and deposition of the


**Table 1.** Parameters values used in simulations [9, 10].

particles influence strongly the spatial distribution of the particles after their injection within tissues. At equilibrium, the temperature field depends significantly on the spatial distribution of the particles. In this analysis, the volume fractions of the particles were considered:

$$\mathbf{O}\_{l}(r) = \frac{\mathbf{C}\_{l}(r)}{\rho\_{\text{MP}}} \tag{7}$$

Q<sup>i</sup>

**Table 2.** The thermal and magnetic characteristics [9, 10].

depend strongly on the particle radius:

*P*[*R*] = *μ*<sup>0</sup> *f π χ*″ [*R*] *H*<sup>0</sup>

relaxation time *τN*[*R*] as function of the particle radius:

*<sup>τ</sup>*[*R*] <sup>=</sup> *<sup>τ</sup><sup>N</sup>*

*τ<sup>B</sup>*

*<sup>k</sup>*

Mass density (kg/m<sup>3</sup>

Anisotropy constant K (kJ/m<sup>3</sup>

H0

*τ*0

(r) = Φ<sup>i</sup>

**Thermal and magnetic characteristics Magnetite Tumor tissue Healthy tissue**

Specific heat capacity (J/kg K) 670 3600 3600 Thermal conductivity (W/mK) 40 0.4692 0.512 Magnetization (kA/m) 446 – –

Frequency f (kHz) 100–650 – – Magnetic field amplitude H (kA/m) 0–15 – –

) 5180 1160 1060

) 9 – –

*P*¯¯ = ∫<sup>0</sup> ∝

is the intensity of the magnetic field, f—frequency of the magnetic field, *μ*<sup>0</sup>

[*R*] <sup>=</sup> <sup>3</sup> *<sup>η</sup> VH* [*R*] \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\_\_*<sup>i</sup> r* 2 \_\_∂ <sup>∂</sup>*r*[*<sup>r</sup>* <sup>2</sup> <sup>∂</sup>*T*\_\_\_*<sup>i</sup>*

*kB <sup>T</sup>* ; *<sup>τ</sup><sup>N</sup>*

is the average relaxation time, *<sup>η</sup>* carrier liquid viscosity, *VH* [R] <sup>=</sup> *<sup>V</sup>*[*R*](<sup>1</sup> <sup>+</sup> \_\_*<sup>δ</sup>*

The size distribution of particles influences strongly the heating rate. For a magnetic system which contains the particles with different sizes, the volumetric heating rate is given by

where the volumetric heating rate released by one particle P[R] and susceptibility *χ*″ [*R*]

the permeability, *Ms* is the saturation magnetization and *kB* = 1.38 ∙ 10−23 J/K is Boltzmann constant. The effective relaxation time contains the Brown relaxation time *τB*[*R*] and the Néel

> *τN* [*R*] + *τ<sup>B</sup>* [*R*]

amic volume of the particles, *δ* is the surfactant layer thickness and K is the anisotropy constant of the magnetic particles. Using the spherical symmetry, Eq. (8) can be written as [17]:

<sup>∂</sup>*<sup>r</sup>* ] <sup>+</sup> *Mi*

*P*[*R*]*g*[*R*]*dR*,

<sup>2</sup> and *<sup>χ</sup>*″ [*R*] <sup>=</sup> *<sup>μ</sup>*<sup>0</sup> *Ms*

[*R*] *τ<sup>B</sup>* [*R*] \_\_\_\_\_\_\_\_\_

[*R*] = *τ*<sup>0</sup>

√ \_\_ \_\_*π* 2

[*T*] = *ρ<sup>i</sup> ci*

exp[ *K V*[*R*] \_\_\_\_\_\_\_\_\_\_\_ *kB <sup>T</sup>* ] \_\_\_\_\_\_\_\_\_\_\_\_\_\_

\_\_\_\_\_\_\_\_\_\_\_ *K V*[*R*] \_\_\_\_\_\_\_\_\_\_\_ *kB T*

.

*R*) 3

<sup>∂</sup>*<sup>t</sup>* (11)

√

∂*T*\_\_\_*<sup>i</sup>*

<sup>2</sup> *<sup>V</sup>*[*R*] \_\_\_\_\_\_\_\_\_\_ 3 *kB T*

(*r*) *P*¯¯ (9)

Modeling of the Temperature Field in the Magnetic Hyperthermia

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311

<sup>2</sup>*<sup>π</sup> <sup>f</sup> <sup>τ</sup>*[*R*] \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

<sup>1</sup> <sup>+</sup> (2*<sup>f</sup> <sup>τ</sup>*[*R*])<sup>2</sup> (10)

 = 4*π* ∙ 10−7 H/m

is the hidrodin-

*ρMNP* is the mass density of the magnetic particles. At the injection site - at the center of the geometry - the maximum value of the volume fraction was Φmax <sup>=</sup> \_\_\_\_ Cmax *ρMNP* :

### **2.2. The temperature field**

The temperature field within malignant and healthy tissues is described by the solutions: *Ti*  = *Ti* (*r*, *t*), (*i* = 1, 2) of the bioheat transfer equation (Pennes equation) in the living tissues [6]:

$$
\rho\_i \mathbf{c}\_i \frac{\partial T\_i}{\partial t} = \nabla \left[ k\_i \left. \nabla T\_i \right| + \rho\_b \mathbf{w}\_b \mathbf{c}\_b \left[ T\_{av} - T\_i \right] + \mathbf{Q}\_{\text{mat}}^i + \mathbf{Q}\_i(r) \right. \tag{8}
$$

with the following thermal characteristics (**Table 2**): ρ<sup>i</sup> —the mass density, c<sup>i</sup> —specific heat capacity, k<sup>i</sup> —the thermal conductivity, ρ<sup>b</sup> —mass density of the blood, cb —specific heat capacity of the blood, Tart—blood temperature, *Q*met *<sup>i</sup>* —metabolic heat production and *Qi* (*r*) (W/m<sup>3</sup> ) power density (volumetric heating rate) dissipated by the magnetic particles within geometric configuration when the magnetic field is applied.

As a result of the spatial distribution of the particles, the total volumetric heating rate *Qi* (*r*) depends on the radial dependent volume fractions of the particles Φ<sup>i</sup> (*r*). When MNP with different sizes are injected within tissues, the volumetric heating rate of the ferrofluid is [7]:


**Table 2.** The thermal and magnetic characteristics [9, 10].

particles influence strongly the spatial distribution of the particles after their injection within tissues. At equilibrium, the temperature field depends significantly on the spatial distribution

> (*r*) <sup>=</sup> Ci (*r*) \_\_\_\_ *ρMNP*

*ρMNP* is the mass density of the magnetic particles. At the injection site - at the center of the

The temperature field within malignant and healthy tissues is described by the solutions:

*wb*

power density (volumetric heating rate) dissipated by the magnetic particles within geometric

As a result of the spatial distribution of the particles, the total volumetric heating rate *Qi*

different sizes are injected within tissues, the volumetric heating rate of the ferrofluid is [7]:

(*r*, *t*), (*i* = 1, 2) of the bioheat transfer equation (Pennes equation) in the living tissues [6]:

*cb*[*Tart* − *Ti*] + *Q*met

—mass density of the blood, cb

*<sup>i</sup>* + *Qi*

*<sup>i</sup>* —metabolic heat production and *Qi*

—the mass density, c<sup>i</sup>

(*r*), (8)

—specific heat capac-

(*r*). When MNP with

—specific heat

(*r*) (W/m<sup>3</sup>

)—

(*r*)

Cmax *ρMNP* :

ε1

K1 = 10−14;

*K*2  = 5 ∙ 10−13;

= 0.1–0.8; ε<sup>2</sup>

= 0.2

(7)

of the particles. In this analysis, the volume fractions of the particles were considered:

Hamaker constant A (J) 3 ∙ 10−21 – 4 ∙ 10−20

(mm) 0.05 – 0.50

) 1000

(J/K) 1.38 ∙ 10−23

Particle radius R (nm) 5 – 30

—healthy tissue porosity

Absolute fluid viscosity *μ* (kg/(s m)) 0.001

—collector zeta potential (mV) −20

—particle zeta potential (mV) −10 to −50

geometry - the maximum value of the volume fraction was Φmax <sup>=</sup> \_\_\_\_

<sup>∂</sup>*<sup>t</sup>* <sup>=</sup> <sup>∇</sup>[*ki* <sup>∇</sup>*Ti*] <sup>+</sup> *<sup>ρ</sup><sup>b</sup>*

depends on the radial dependent volume fractions of the particles Φ<sup>i</sup>

Φ<sup>i</sup>

**Table 1.** Parameters values used in simulations [9, 10].

∂*T*\_\_\_*<sup>i</sup>*

with the following thermal characteristics (**Table 2**): ρ<sup>i</sup>

—the thermal conductivity, ρ<sup>b</sup>

configuration when the magnetic field is applied.

ity of the blood, Tart—blood temperature, *Q*met

**2.2. The temperature field**

Collector diameter dc

(*i* = 1, 2)

310 Numerical Simulations in Engineering and Science

—malignant tissue porosity; *ε*<sup>2</sup>

(*i* = 1, 2)

—malignant tissue permeability;

—healthy tissue permeability

Water mass density *ρ<sup>W</sup>* (kg/m3

Boltzmann coefficient *kb*

Tissue porosity: *ε<sup>i</sup>*

Permeability *Ki*

*ε*1

*K*1

*K*2

*ζp*

*ζc*

*ρ<sup>i</sup> ci*

*Ti*  = *Ti*

capacity, k<sup>i</sup>

$$\mathbf{Q}\_{\!\!\!\!-1}(\mathbf{r}) = \mathbf{Q}\_{\!\!\!\!-1}(\mathbf{r}) \,\mathbf{\tilde{P}} \tag{9}$$

The size distribution of particles influences strongly the heating rate. For a magnetic system which contains the particles with different sizes, the volumetric heating rate is given by

$$\overline{P} = \int\_0^\infty P[R]g[R]dR,$$

where the volumetric heating rate released by one particle P[R] and susceptibility *χ*″ [*R*] depend strongly on the particle radius:

$$P\{\text{R}\} = \mu\_0 f \pi \,\chi^{'}\{\text{R}\} \, H\_0^2 \text{ and } \chi^{'}\{\text{R}\} = \frac{\mu\_0 M\_\*^4 \text{V} \text{[R}\text{]}}{3 \, k\_\text{s} \, T} \frac{2 \pi f \tau \text{[R}\text{]}}{1 + (2 \pi f \tau \text{[R}\text{]})^2} \tag{10}$$

H0 is the intensity of the magnetic field, f—frequency of the magnetic field, *μ*<sup>0</sup>  = 4*π* ∙ 10−7 H/m the permeability, *Ms* is the saturation magnetization and *kB* = 1.38 ∙ 10−23 J/K is Boltzmann constant. The effective relaxation time contains the Brown relaxation time *τB*[*R*] and the Néel relaxation time *τN*[*R*] as function of the particle radius:

$$\tau[R] = \frac{\tau\_{\tiny\tiny R}[R]\,\tau\_{\tiny\tiny R}[R]}{\tau\_{\tiny\tiny\tiny R}[R] + \tau\_{\tiny\tiny\textrm{R}}[R]\,\tau\_{\tiny\textrm{R}}[R]}$$

$$\tau\_{\rm{B}}[\rm{R}] = \frac{3\,\eta\,V\_{\rm{H}}\,\text{[R]}}{k\_{\rm{B}}\,\text{T}};\tau\_{\rm{N}}[\rm{R}] = \tau\_{\rm{0}}\frac{\sqrt{\pi}}{2}\frac{\exp\left[\frac{\text{K}\,\text{V}\,\text{[R]}}{k\_{\rm{x}}\,\text{T}}\right]}{\sqrt{\frac{\text{K}\,\text{V}\,\text{[R]}}{k\_{\rm{x}}\,\text{T}}}}.$$

*τ*0 is the average relaxation time, *<sup>η</sup>* carrier liquid viscosity, *VH* [R] <sup>=</sup> *<sup>V</sup>*[*R*](<sup>1</sup> <sup>+</sup> \_\_*<sup>δ</sup> R*) 3 is the hidrodinamic volume of the particles, *δ* is the surfactant layer thickness and K is the anisotropy constant of the magnetic particles. Using the spherical symmetry, Eq. (8) can be written as [17]:

$$\frac{k\_i}{r^2} \frac{\partial}{\partial r} \left[ r^2 \frac{\partial T\_i}{\partial r} \right] + M\_i[T] = \rho\_i c\_i \frac{\partial T\_i}{\partial t} \tag{11}$$

with M*<sup>i</sup>* [*T*] <sup>=</sup> ωb <sup>i</sup> *<sup>ρ</sup><sup>b</sup> <sup>c</sup> <sup>b</sup>*[*Te i* (r) <sup>−</sup> *Ti* ]. For i = 1, 2, the expressions *Te i* (r) are:

$$T\_\epsilon^1(\mathbf{r}) = \mathbf{T}\_\mathsf{B}^1 + \frac{Q\_\mathbf{i}(r)}{\omega\_\mathsf{b}^1 c\_\mathsf{b} \rho\_\mathsf{b}} \text{ and } T\_\epsilon^2(\mathbf{r}) = \mathbf{T}\_\mathsf{B}^2 + \frac{Q\_\mathbf{i}(r)}{\omega\_\mathsf{b}^2 c\_\mathsf{b} \rho\_\mathsf{b}}$$

where

$$\mathbf{T\_{B}^{1}} = \mathbf{T\_{b}} + \frac{Q\_{\rm mt}^{1}}{\alpha\_{\rm b}^{1}c\_{\rm b}\rho\_{\rm b}} \text{ and } \mathbf{T\_{B}^{2}} = \mathbf{T\_{b}} + \frac{Q\_{\rm mt}^{2}}{\alpha\_{\rm b}^{2}c\_{\rm b}\rho\_{\rm b}}.$$

Eq. (8) was solved in **Appendix 2** at the thermal equilibrium. The general solutions are obtained:

$$T\_1(\mathbf{r}) = \mathbf{c}\_1 \frac{\cosh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} + \mathbf{c}\_2 \frac{\sinh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} + \mathbf{v}\_1 \frac{\cosh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} + \mathbf{v}\_2 \frac{\sinh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} \tag{A2.3}$$

*<sup>T</sup>*<sup>1</sup>

*T*2

*k*<sup>1</sup> [

**3. Results and discussions**

the magnetic field parameters: (H<sup>0</sup>

mathematical conditions.

infusion rate Qv

(r) = c3

(r) = c2

cosh(β<sup>2</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> c4

sinh(β<sup>1</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>v</sup><sup>1</sup>

∂*T*\_\_\_1 <sup>∂</sup>*<sup>r</sup>* ]*<sup>R</sup>*<sup>1</sup>

the ablation of the whole tumor according with Hergt condition: *H*<sup>0</sup> *f* < 5 ∙ 10<sup>9</sup>

dence of the pressure for the same values of the parameter Qv

cosh(β<sup>1</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>v</sup><sup>2</sup>

sinh(β<sup>2</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>v</sup><sup>3</sup>

(ii) Dirichlet boundary condition was considered on the external surface of the healthy tissue:

*T*2[r = R2] = 37° C (13)

(iii) The Newman boundary conditions are considered at malignant—healthy tissue interface. The heat flux coming from the malignant tissue is completely received by the healthy region. The continuity condition of the heat fluxes is imposed at tumor—healthy region interface:

> = *k*<sup>2</sup> [ <sup>∂</sup>*T*\_\_\_2 <sup>∂</sup>*<sup>r</sup>* ]*<sup>R</sup>*<sup>1</sup>

*T*1[*r* = *R*1] = *T*2[*r* = *R*1] (15)

In this analysis, the magnetite system with sizes in the range (5–30) nm was considered. The temperature field within a liver tissue was computed for different values of the magnetic field parameters. The malignant and healthy tissues are two concentric domains having the diameter of 20 mm and 100 mm, respectively (**Figure 1**). The temperature values depend strongly on the particle size and magnetic field parameters (frequency and amplitude). The values of

(1), (3), (5) and (8) were solved in a numerical model using the finite element method (FEM). Their numerical solutions were compared by the previous analytical solutions in the same

**Figure 2(a)** shows the radial dependence of the particle velocity for values of the ferrofluid

ences significantly the particles velocities within tissues. **Figure 2(b)** shows the radial depen-

and (4.2), the pressure decreases with the distance from IS. The pressure developed within geometry as result of ferrofluid infusion depends strongly on the Qv. Higher values of Qv determines higher values of pressure and faster movements of the particles within tissues. As result of the convection process, the particles move on larger distances or remain in the small vicinity of the IS (at small radial distances). The parameter Qv and implicitly the pressure generated within tissue by the ferrofluid infusion influences strongly the convection—diffusion deposition processes of the particles and implicitly the spatial distribution of the particles.

decreases with the distance from IS according with the relation (2). The parameter Qv

in the range of 5–30 μl/min. The velocity of the particles on radial direction,

sinh(β1 r) \_\_\_\_\_\_\_\_\_\_\_\_\_ *r*

Modeling of the Temperature Field in the Magnetic Hyperthermia

sinh(β<sup>2</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>*

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

(12)

313

(14)

Am−1 s−1 [7]. Eqs.

. In agreement with relations (4.1)

influ-

cosh(β<sup>2</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>v</sup><sup>4</sup>

and f) verify the criterion of exposure safe and tolerable for

and

$$T\_2(\mathbf{r}) = \mathbf{c}\_3 \frac{\cosh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} + \mathbf{c}\_4 \frac{\sinh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} + \mathbf{v}\_3 \frac{\cosh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} + \mathbf{v}\_4 \frac{\sinh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} \tag{A2.4}$$

The expressions v<sup>1</sup> , v<sup>2</sup> , v<sup>3</sup> and v<sup>4</sup> are:

$$\begin{aligned} \mathbf{v}\_{1} &= \frac{1}{\beta\_{1}} \mathbf{f} \text{ } \left[ \mathbf{a}\_{1} + \mathbf{b}\_{1} \, \boldsymbol{\Phi}\_{1}(\boldsymbol{r}) \right] \sinh(\beta\_{1} \mathbf{r}) \text{d}\mathbf{r} \text{ and } \mathbf{v}\_{2} = -\frac{1}{\beta\_{1}} \int \mathbf{r} \left[ \mathbf{a}\_{1} + \mathbf{b}\_{1} \, \boldsymbol{\Phi}\_{1}(\boldsymbol{r}) \right] \cosh(\beta\_{1} \mathbf{r}) \text{d}\mathbf{r} \\\\ \mathbf{v}\_{3} &= \frac{1}{\beta\_{2}} \int \mathbf{r} \left[ \mathbf{a}\_{2} + \mathbf{b}\_{2} \, \boldsymbol{\Phi}\_{2}(\boldsymbol{r}) \right] \sinh(\beta\_{2} \mathbf{r}) \text{d}\mathbf{r} \text{ and } \mathbf{v}\_{4} = -\frac{1}{\beta\_{2}} \int \mathbf{r} \left[ \mathbf{a}\_{2} + \mathbf{b}\_{2} \, \boldsymbol{\Phi}\_{2}(\boldsymbol{r}) \right] \cosh(\beta\_{2} \mathbf{r}) \text{d}\mathbf{r} \end{aligned}$$

with the following notations:

$$\begin{aligned} a\_1 &= \beta\_1^2 \,\mathrm{T}\_{\mathrm{b}}^1 \text{ and } a\_2 = \beta\_2^2 \,\mathrm{T}\_{\mathrm{b}}^2 \\\\ \mathbf{b}\_1 &= \frac{\beta\_1^2 \,\mathrm{P}}{\omega\_{\mathrm{b}}^1 \,c\_{\mathrm{b}} \,\rho\_{\mathrm{b}}} \text{ and } \mathbf{b}\_2 = \frac{\beta\_2^2 \,\mathrm{P}}{\omega\_{\mathrm{b}}^2 \,c\_{\mathrm{b}} \,\rho\_{\mathrm{b}}} \\\\ \beta\_1^2 &= \frac{\omega\_{\mathrm{b}}^1 c\_{\mathrm{b}} \,\rho\_{\mathrm{b}}}{k\_1} \beta\_2^2 = \frac{\omega\_{\mathrm{b}}^2 c\_{\mathrm{b}} \,\rho\_{\mathrm{b}}}{k\_2} \end{aligned}$$

The integration constants: c<sup>1</sup> , *c*2 , *c*3 , *c*4 were computed from the following boundary conditions. **Boundary conditions:**

(i) The temperature *T*<sup>1</sup> is finite at the center (*r*→ 0) of the geometric structure (**Figure 1**). As a result, the constant *c*<sup>1</sup> is zero. Therefore, the temperatures: *T*<sup>1</sup> (*r*) and *T*<sup>2</sup> (*r*) are (**Appendix 2**):

### Modeling of the Temperature Field in the Magnetic Hyperthermia http://dx.doi.org/10.5772/intechopen.71364 313

$$\begin{aligned} T\_1(\mathbf{r}) &= \mathbf{c}\_2 \frac{\sinh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} + \mathbf{v}\_1 \frac{\cosh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} + \mathbf{v}\_2 \frac{\sinh(\boldsymbol{\beta}\_1 \cdot \mathbf{r})}{r} \\ T\_2(\mathbf{r}) &= \mathbf{c}\_3 \frac{\cosh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} + \mathbf{c}\_4 \frac{\sinh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} + \mathbf{v}\_3 \frac{\cosh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} + \mathbf{v}\_4 \frac{\sinh(\boldsymbol{\beta}\_2 \cdot \mathbf{r})}{r} \end{aligned} \tag{12}$$

(ii) Dirichlet boundary condition was considered on the external surface of the healthy tissue:

$$T\_2[\mathbf{r} = \mathbf{R}\_2] = \mathcal{ST}^+\mathbf{C} \tag{13}$$

(iii) The Newman boundary conditions are considered at malignant—healthy tissue interface. The heat flux coming from the malignant tissue is completely received by the healthy region. The continuity condition of the heat fluxes is imposed at tumor—healthy region interface:

$$\left.k\_1\left[\frac{\partial T\_1}{\partial r}\right]\_{R\_1}\right| = \left.k\_2\left[\frac{\partial T\_2}{\partial r}\right]\_{R\_1}\right|\tag{14}$$

$$T\_\mathbf{i}[r = R\_\mathbf{i}] = \ \ ^T\_\mathbf{i}[r = R\_\mathbf{i}] \tag{15}$$

## **3. Results and discussions**

with M*<sup>i</sup>*

where

obtained:

and

*T*<sup>1</sup>

*T*<sup>2</sup>

The expressions v<sup>1</sup>

β1

β2

with the following notations:

<sup>v</sup><sup>1</sup> <sup>=</sup> \_\_1

<sup>v</sup><sup>3</sup> <sup>=</sup> \_\_1

[*T*] <sup>=</sup> ωb

<sup>i</sup> *<sup>ρ</sup><sup>b</sup> <sup>c</sup> <sup>b</sup>*[*Te i* (r) <sup>−</sup> *Ti*

312 Numerical Simulations in Engineering and Science

*Te*

TB

(r) = c<sup>1</sup>

(r) = c<sup>3</sup>

and v<sup>4</sup>

, v<sup>2</sup> , v<sup>3</sup>

∫ r [a1 + b<sup>1</sup> Φ<sup>1</sup>

∫ r [a2 + b<sup>2</sup> Φ<sup>2</sup>

*a*<sup>1</sup> = *β*<sup>1</sup>

<sup>b</sup><sup>1</sup> <sup>=</sup> *<sup>β</sup>*<sup>1</sup>

*β*<sup>1</sup>

The integration constants: c<sup>1</sup>

**Boundary conditions:**

(i) The temperature *T*<sup>1</sup>

a result, the constant *c*<sup>1</sup>

]. For i = 1, 2, the expressions *Te*

*Q*met 1 \_\_\_\_\_\_ ωb <sup>1</sup> *cb ρ<sup>b</sup>*

and *Te*

and TB

Eq. (8) was solved in **Appendix 2** at the thermal equilibrium. The general solutions are

sinh(β<sup>1</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>v</sup><sup>1</sup>

sinh(β<sup>2</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>v</sup><sup>3</sup>

(*r*)] sinh(β<sup>1</sup> <sup>r</sup>)dr and <sup>v</sup><sup>2</sup> <sup>=</sup> <sup>−</sup> \_\_1

(*r*)] sinh(β<sup>2</sup> <sup>r</sup>)dr and <sup>v</sup><sup>4</sup> <sup>=</sup> <sup>−</sup> \_\_1

<sup>2</sup> TB

<sup>2</sup> *<sup>P</sup>*¯ \_\_\_\_\_\_

ωb <sup>1</sup> *cb ρ<sup>b</sup>*

<sup>2</sup> <sup>=</sup> ωb <sup>1</sup> *cb ρ* \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_*<sup>b</sup> k*1 ; *β*2 <sup>2</sup> <sup>=</sup> ωb <sup>2</sup> *cb ρ* \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_*<sup>b</sup> k*2

is zero. Therefore, the temperatures: *T*<sup>1</sup>

, *c*2 , *c*3 , *c*4 <sup>1</sup> and *a*<sup>2</sup> = *β*<sup>2</sup>

and <sup>b</sup><sup>2</sup> <sup>=</sup> *<sup>β</sup>*<sup>2</sup>

1(r) = TB

<sup>1</sup> = T<sup>b</sup> +

cosh(β<sup>1</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>c</sup><sup>2</sup>

cosh(β<sup>2</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>c</sup><sup>4</sup>

are:

<sup>1</sup> + *Q*1 (*r*) \_\_\_\_\_\_\_\_\_\_ ωb <sup>1</sup> *cb ρ<sup>b</sup>*

*i* (r) are:

2(r) = TB

<sup>2</sup> = T<sup>b</sup> +

<sup>2</sup> + *Q*2 (*r*) \_\_\_\_\_\_\_\_\_\_ *ω*<sup>2</sup> *<sup>b</sup> cb ρ<sup>b</sup>*

*Q*met 2 \_\_\_\_\_\_ ωb <sup>2</sup> *cb ρ<sup>b</sup>* .

cosh(β<sup>1</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>v</sup><sup>2</sup>

cosh(β<sup>2</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* <sup>+</sup> <sup>v</sup><sup>4</sup>

∫ r [a1 + b<sup>1</sup> Φ<sup>1</sup>

∫ r [a2 + b<sup>2</sup> Φ<sup>2</sup>

were computed from the following boundary conditions.

(*r*) and *T*<sup>2</sup>

β1

β2

<sup>2</sup> TB 2

<sup>2</sup> *<sup>P</sup>*¯ \_\_\_\_\_\_ ωb <sup>2</sup> *cb ρ<sup>b</sup>*

is finite at the center (*r*→ 0) of the geometric structure (**Figure 1**). As

sinh(β<sup>1</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* (A2.3)

sinh(β<sup>2</sup> <sup>r</sup>) \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>r</sup>* (A2.4)

(*r*)] cosh(β<sup>1</sup> r)dr

(*r*)] cosh(β<sup>2</sup> r)dr

(*r*) are (**Appendix 2**):

In this analysis, the magnetite system with sizes in the range (5–30) nm was considered. The temperature field within a liver tissue was computed for different values of the magnetic field parameters. The malignant and healthy tissues are two concentric domains having the diameter of 20 mm and 100 mm, respectively (**Figure 1**). The temperature values depend strongly on the particle size and magnetic field parameters (frequency and amplitude). The values of the magnetic field parameters: (H<sup>0</sup> and f) verify the criterion of exposure safe and tolerable for the ablation of the whole tumor according with Hergt condition: *H*<sup>0</sup> *f* < 5 ∙ 10<sup>9</sup> Am−1 s−1 [7]. Eqs. (1), (3), (5) and (8) were solved in a numerical model using the finite element method (FEM). Their numerical solutions were compared by the previous analytical solutions in the same mathematical conditions.

**Figure 2(a)** shows the radial dependence of the particle velocity for values of the ferrofluid infusion rate Qv in the range of 5–30 μl/min. The velocity of the particles on radial direction, decreases with the distance from IS according with the relation (2). The parameter Qv influences significantly the particles velocities within tissues. **Figure 2(b)** shows the radial dependence of the pressure for the same values of the parameter Qv . In agreement with relations (4.1) and (4.2), the pressure decreases with the distance from IS. The pressure developed within geometry as result of ferrofluid infusion depends strongly on the Qv. Higher values of Qv determines higher values of pressure and faster movements of the particles within tissues. As result of the convection process, the particles move on larger distances or remain in the small vicinity of the IS (at small radial distances). The parameter Qv and implicitly the pressure generated within tissue by the ferrofluid infusion influences strongly the convection—diffusion deposition processes of the particles and implicitly the spatial distribution of the particles.

The evolution with the parameter Qv

remain deposited on the solid matrix.

of the volume fraction of the particles Φ<sup>1</sup>

The values of the magnetic field parameters: (H<sup>0</sup>

*f* 1

velocities (high values of Qv

the particle zeta potential *ζ<sup>p</sup>*

The coefficient *k*

*ζp*

tissue.

of the deposition rate coefficient of the particles *k*

Modeling of the Temperature Field in the Magnetic Hyperthermia

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

) as a result of the high pressure gradient have no capability to

. **Figure 5(a)** shows the evolution with radial distance from IS

(*r*) for different values of the particle zeta potential

and f) are essential in the optimization of

studied for different porosities of the malignant tissue, in order to understand the influence of this parameter on the deposition process of the particles on the solid porous matrix (**Figure 4**).

decreases with the increase of the value of the parameter Qv

tion of the particles decreases with the increase of the tissue porosity. The particles with high

The repulsive electrostatic double layer forces influences the particle deposition process and implicitly the spatial distribution of the particles. Temperature field depends strongly on

 =  − 10 to − 40 mV. As a result of the strong repulsive electrostatic double layer forces, a number of the particles are deposited in the solid structure of tissue. This effect influences significantly the spatial distribution of the temperature (on radial direction) as **Figure 5(b)** shows. As a consequence, the temperature gradients become smaller in the case of smaller repulsive electrostatic double layer forces. The repulsive electrostatic interactions (as a result of the repulsive electrostatic double layer (EDL) forces) influence strongly the mass concentration of the par-

ferrofluid (as liquid medium) due to the ionic conditions measured by pH and ionic strength.

the Magnetic Hyperthermia therapy. In the following, the increase of the temperature on the

**Figure 4.** The evolution with the parameter Qv of deposition rate coefficient of the particles, *k f* 1 within the malignant

ticles and the spatial temperature field. The particle zeta potential *ζ<sup>p</sup>*

radial direction with the frequency of the magnetic field was followed.

*f* 1 was 315

. Also the deposi-

can be controlled in the

**Figure 2.** (a) The radial velocity of the magnetite particles within tissues; (b) the pressure on radial direction within malignant tissue.

The radial distribution of the particles and temperature field were analyzed for different values of the parameter Qv in the range 10–40 μl/min. **Figure 3(a)** shows the evolution with distance from IS of the volume fraction of the particles Φ<sup>1</sup> (*r*) within tumor for different values of Qv . The value of the concentration at IS was Cmax = 10 mg/cm<sup>3</sup> . Consequently, the maxim value of the volume fraction at IS was Φmax = 1.93 ∙ 10−3. Variation of the parameter Qv determines different spatial distributions of the particles within the malignant tissue which influences strongly the temperature field (**Figure 3(b)**). Practically, the temperature values within the malignant tissue can be controlled in the therapeutic range (42–46)°C by using an optimum value of Qv during the ferrofluid infusion process.

**Figure 3.** The influence of the parameter Qv on the radial dependent volume fraction of the particles (a) and the temperature field (b).

The evolution with the parameter Qv of the deposition rate coefficient of the particles *k f* 1 was studied for different porosities of the malignant tissue, in order to understand the influence of this parameter on the deposition process of the particles on the solid porous matrix (**Figure 4**). The coefficient *k f* 1 decreases with the increase of the value of the parameter Qv . Also the deposition of the particles decreases with the increase of the tissue porosity. The particles with high velocities (high values of Qv ) as a result of the high pressure gradient have no capability to remain deposited on the solid matrix.

The repulsive electrostatic double layer forces influences the particle deposition process and implicitly the spatial distribution of the particles. Temperature field depends strongly on the particle zeta potential *ζ<sup>p</sup>* . **Figure 5(a)** shows the evolution with radial distance from IS of the volume fraction of the particles Φ<sup>1</sup> (*r*) for different values of the particle zeta potential *ζp*  =  − 10 to − 40 mV. As a result of the strong repulsive electrostatic double layer forces, a number of the particles are deposited in the solid structure of tissue. This effect influences significantly the spatial distribution of the temperature (on radial direction) as **Figure 5(b)** shows. As a consequence, the temperature gradients become smaller in the case of smaller repulsive electrostatic double layer forces. The repulsive electrostatic interactions (as a result of the repulsive electrostatic double layer (EDL) forces) influence strongly the mass concentration of the particles and the spatial temperature field. The particle zeta potential *ζ<sup>p</sup>* can be controlled in the ferrofluid (as liquid medium) due to the ionic conditions measured by pH and ionic strength.

The radial distribution of the particles and temperature field were analyzed for different val-

**Figure 2.** (a) The radial velocity of the magnetite particles within tissues; (b) the pressure on radial direction within

different spatial distributions of the particles within the malignant tissue which influences strongly the temperature field (**Figure 3(b)**). Practically, the temperature values within the malignant tissue can be controlled in the therapeutic range (42–46)°C by using an optimum

**Figure 3.** The influence of the parameter Qv on the radial dependent volume fraction of the particles (a) and the

of the volume fraction at IS was Φmax = 1.93 ∙ 10−3. Variation of the parameter Qv

in the range 10–40 μl/min. **Figure 3(a)** shows the evolution with dis-

(*r*) within tumor for different values of

. Consequently, the maxim value

determines

ues of the parameter Qv

314 Numerical Simulations in Engineering and Science

malignant tissue.

Qv

value of Qv

temperature field (b).

tance from IS of the volume fraction of the particles Φ<sup>1</sup>

. The value of the concentration at IS was Cmax = 10 mg/cm<sup>3</sup>

during the ferrofluid infusion process.

The values of the magnetic field parameters: (H<sup>0</sup> and f) are essential in the optimization of the Magnetic Hyperthermia therapy. In the following, the increase of the temperature on the radial direction with the frequency of the magnetic field was followed.

**Figure 4.** The evolution with the parameter Qv of deposition rate coefficient of the particles, *k f* 1 within the malignant tissue.

**Figure 5.** The influence of the parameter—zeta potential ξp on the radial dependent volume fraction of the particles (a) and the temperature field (b).

**Figure 6(a)** and **(b)** show a 3D and 2D view of the radial temperature field within the malignant tissue for different values of the frequency of the magnetic field. It was considered a small value of both Qv and the tissue porosity in order to analyze a temperature field with strong non-uniformity (and implicitly high thermal gradients).

The analytical temperature was compared with the numerical results given by the finite element method (FEM) in Comsol Multiphysics using the same parameters in the same mathematical conditions. Good agreements were found between the predictions of the analytical

**Figure 7.** The isothermal surfaces for different parameters. (a) Isothermal surfaces for different values of the parameter Qv and frequency of the magnetic field f and (b) isothermal surfaces for different values of the tissue porosity and

Modeling of the Temperature Field in the Magnetic Hyperthermia

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The simulations allow (i) the optimization of the main parameters which influences strongly the heating of the tumor in the therapeutic temperature range and (ii) provide better tempera-

The model developed in this paper analyzes the essential role of the ferrofluid infusion rate in the radial MNP distribution after their injection within a malignant tissue. Analytical correlations between the following parameters: (i) the particle velocity, (ii) the pressure developed in geometry during the ferrofluid infusion and (iii) the particle concentration were done in order to understand and predicts the temperature field within tissues when an external magnetic field is applied. The temperature field is concentrated within the malignant tissue. The tem-

The thermal energy deposited within the malignant tissue provides from the MNP distributed as result of convection-diffusion-deposition of the particles after their injection inside tissue. The ferrofluid infusion rate influences significantly the radial distribution of the particles

The temperature field within the malignant tissues can be controlled by the control of the ferrofluid infusion rate Qv during the infusion process. The particles having higher velocity moves

perature on the tumor border (approximately 38–39°C) not affects the healthy tissue.

model and numerical results.

frequency of the magnetic field f.

**4. Conclusion**

ture control through treatment preplanning.

and consequently the temperature field.

**Figure 7(a)** shows the values of the main parameters Qv and f which determines the same temperature on the radial direction. **Figure 7(b)** shows the isothermal surfaces for different values of values of the main parameters Qv and f.

**Figure 6.** The evolution with the frequency of the magnetic field of the temperature field on radial direction Qv = 10 μl/min; ɛ<sup>1</sup> = 0.2 and ξ<sup>p</sup> = −30 mV. (a) 3D view and (b) 2D view.

**Figure 7.** The isothermal surfaces for different parameters. (a) Isothermal surfaces for different values of the parameter Qv and frequency of the magnetic field f and (b) isothermal surfaces for different values of the tissue porosity and frequency of the magnetic field f.

The analytical temperature was compared with the numerical results given by the finite element method (FEM) in Comsol Multiphysics using the same parameters in the same mathematical conditions. Good agreements were found between the predictions of the analytical model and numerical results.

The simulations allow (i) the optimization of the main parameters which influences strongly the heating of the tumor in the therapeutic temperature range and (ii) provide better temperature control through treatment preplanning.
