**3. Advanced computer modeling in breast cancer hyperthermia treatment**

In this section, we present a computer modeling for microwave high hyperthermia in breast cancer treatment. Computational electromagnetic (CEM) or electromagnetic modeling employs numerical methods to describe propagation of electromagnetic waves. It typically involves the formulation of discrete solutions using computationally efficient approximations to Maxwell's equations. There are three techniques of CEM: the finitedifference time-domain (FDTD), the method of moments (MOM), and the finite element method (FEM), which has been extensively used in simulations of cardiac and hepatic radiofrequency (RF) ablation [26]. A FEM model was used in this work because it can provide users with quick, accurate solutions to multiple systems of differential equations and therefore, they are well suited to solve heat transfer problems like ablation [27]. Numerous MWA antenna designs specically targeted for MWA cardiac and hepatic applications have been reported [20-24], but they have not been used to treat breast cancer. These designs have been focused largely on thin, coaxial-based interstitial antennas [28], which are minimally invasive and capable of delivering a large amount of electromagnetic power. These antennas can usually be classified as one of three types (dipole, slot, or monopole) based on their physical features and radiation properties [29]. On the other hand, several researchers are investigating non-invasive microwave hyperthermia for treatment of breast cancer [30].

#### **3.1. Equations**

The frequency-dependent reflection coefficient can be expressed logarithmically as:

$$
\Gamma(f) = 10 \cdot \log\_{10} \left( \frac{p\_r(f)}{p\_{ln}} \right) [dB] \tag{1}
$$

where, Pin is the input power and Pr indicates the reflected power (W). SAR represents the amount of time average power deposited per unit mass of tissue (W/kg) at any position. It can be expressed mathematically as:

$$SAR = \frac{\sigma}{\rho} \left| E \right|^2 = \left[ W / \text{Kg} \right] \tag{2}$$

where, σ is tissue conductivity (S/m), ρ is tissue density (kg/m3) and E is the electric field [56]. The SAR takes a value proportional to the square of the electric field generated around the antenna and is equivalent to the heating source created by the electric field in the tissue. The SAR pattern of an antenna causes the tissue temperature to rise, but does not determine the final tissue temperature distribution directly. The tissue temperature increment results from both power and time. MW heating thermal effects can be roughly described by Pennes' Bioheat equation [31]:

$$\nabla \cdot \left( -k \nabla T \right) = \rho\_b \mathcal{C}\_b \omega\_b \left( T\_b - T \right) + Q\_{\text{met}} + Q\_{\text{ext}} \tag{3}$$

where k is the tissue thermal conductivity (W/m°K), ρb is the blood density (Kg/m3), Cb is the blood specific heat (J/Kg°K), ωb is the blood perfusion rate (1/s). Tb is the temperature of the blood and T is the final temperature. Qmet is the heat source from metabolism and Qext an external heat source. The major physical phenomena considered in the equation are microwave heating and tissue heat conduction. The temperature of the blood is approximated as the core temperature of the body. Moreover, in ex vivo samples, ωb and Qmet can be neglected since no perfusion or metabolism exists. The external heat source is equal to the resistive heat generated by the electromagnetic field.

#### **3.2. Material properties**

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effects or complications.

breast cancer [30].

**3.1. Equations** 

can be expressed mathematically as:

vicinity of vascular structures are limitations of current devices. An ideal ablative technology would ensure complete destruction of all malignant cells with no significant side

**3. Advanced computer modeling in breast cancer hyperthermia treatment** 

In this section, we present a computer modeling for microwave high hyperthermia in breast cancer treatment. Computational electromagnetic (CEM) or electromagnetic modeling employs numerical methods to describe propagation of electromagnetic waves. It typically involves the formulation of discrete solutions using computationally efficient approximations to Maxwell's equations. There are three techniques of CEM: the finitedifference time-domain (FDTD), the method of moments (MOM), and the finite element method (FEM), which has been extensively used in simulations of cardiac and hepatic radiofrequency (RF) ablation [26]. A FEM model was used in this work because it can provide users with quick, accurate solutions to multiple systems of differential equations and therefore, they are well suited to solve heat transfer problems like ablation [27]. Numerous MWA antenna designs specically targeted for MWA cardiac and hepatic applications have been reported [20-24], but they have not been used to treat breast cancer. These designs have been focused largely on thin, coaxial-based interstitial antennas [28], which are minimally invasive and capable of delivering a large amount of electromagnetic power. These antennas can usually be classified as one of three types (dipole, slot, or monopole) based on their physical features and radiation properties [29]. On the other hand, several researchers are investigating non-invasive microwave hyperthermia for treatment of

The frequency-dependent reflection coefficient can be expressed logarithmically as:

Γ(�) = 10 ∙ log�� �

where, Pin is the input power and Pr indicates the reflected power (W). SAR represents the amount of time average power deposited per unit mass of tissue (W/kg) at any position. It

> <sup>2</sup> *SAR E W Kg* /

where, σ is tissue conductivity (S/m), ρ is tissue density (kg/m3) and E is the electric field [56]. The SAR takes a value proportional to the square of the electric field generated around the antenna and is equivalent to the heating source created by the electric field in the tissue. The SAR pattern of an antenna causes the tissue temperature to rise, but does not determine the final tissue temperature distribution directly. The tissue temperature increment results

���(�) ���

� ���� (1)

(2)

The computer antenna model used in this work is based on a 50Ω UT-085 semirigid coaxial cable. The entire outer conductor is copper, in which a small ring slot of width is cut close to the short-circuited distal tip of the antenna to allow electromagnetic wave propagation into the tissue. The inner conductor is made from silver-plated copper wire (SPCW) and the coaxial dielectric is a low-loss polytetrauoroethylene (PTFE). The length of the antenna also affects the power reection and shape of the SAR pattern. Furthermore, the antenna is encased in a PTFE catheter to prevent adhesion of the antenna to desiccated ablated tissue. Dimensions and thermal properties of the materials and breast tissue, which were taken from the literature [32], are listed in Table 5 and 6.




High Temperature Hyperthermia in Breast Cancer Treatment 95

A finite element method computer models were developed using COMSOL Multiphysics 4.0 commercial software. One of the models assumed that the coaxial slot antenna was immersed only in homogeneous breast tissue; the other model assumed that the antenna was immersed only in breast cancer. The coaxial slot antenna exhibits rotational symmetry around the longitudinal axis; therefore axisymmetric models, which minimized the computation time, were used. The inner and outer conductors of the antenna were modeled using perfect electric conductor boundary conditions and boundaries along the z axis were

All boundaries of conductors were set to perfect electric conductor (PEC). Boundaries along the *z* axis were set with axial symmetry and all other boundaries were set to low reflection boundaries. Figure 2 shows the geometry of the antenna model with details near its slot; since the model is axisymmetric, only a half of the antenna geometry structure is shown [33].

**Figure 2.** Axisymmetric model in the vicinity of the tip of the coaxial slot antenna. The vertical axis (z) corresponds to the longitudinal axis of the antenna; while the horizontal axis (r) corresponds to radial

set with axial symmetry.

direction. All units are in meters.


Figure 1 shows the axial schematics of each section of the antenna and the interior diameters.

**Figure 1.** Cross section and axial schematic of the coaxial slot antenna.

A finite element method computer models were developed using COMSOL Multiphysics 4.0 commercial software. One of the models assumed that the coaxial slot antenna was immersed only in homogeneous breast tissue; the other model assumed that the antenna was immersed only in breast cancer. The coaxial slot antenna exhibits rotational symmetry around the longitudinal axis; therefore axisymmetric models, which minimized the computation time, were used. The inner and outer conductors of the antenna were modeled using perfect electric conductor boundary conditions and boundaries along the z axis were set with axial symmetry.

94 Hyperthermia

diameters.

Material Relative permittivity

Catheter 2.60 Breast tissue 5.14 Tumor 57

Figure 1 shows the axial schematics of each section of the antenna and the interior

Inner dielectric oft he coaxial cable 2.03

**Table 6.** Relative permittivity for the materials and tissue.

**Figure 1.** Cross section and axial schematic of the coaxial slot antenna.

All boundaries of conductors were set to perfect electric conductor (PEC). Boundaries along the *z* axis were set with axial symmetry and all other boundaries were set to low reflection boundaries. Figure 2 shows the geometry of the antenna model with details near its slot; since the model is axisymmetric, only a half of the antenna geometry structure is shown [33].

**Figure 2.** Axisymmetric model in the vicinity of the tip of the coaxial slot antenna. The vertical axis (z) corresponds to the longitudinal axis of the antenna; while the horizontal axis (r) corresponds to radial direction. All units are in meters.

### **3.3. Results**

Figure 3 shows the temperature distribution in normal adipose-dominated tissue [34]. The reflection coefficient calculated for the frequency at 2.45 GHz was -2.82 dB, the maximum temperature was 116.03 ºC, and the ablation zone radius was 53 mm. The isotherm was considered at 60 ºC because ablation is produced above this temperature [35]. Figure 4 shows the temperature distribution in breast cancer tissue. The reflection coefficient calculated for the frequency at 2.45 GHz was -6.38 dB, the maximum temperature was 125.96 ºC, and the ablation zone radius was 92 mm.

High Temperature Hyperthermia in Breast Cancer Treatment 97

**Figure 4.** Temperature distribution of breast cancer tissue at a microwave power output of 10 W. The isotherm at 60 ºC is highlighted. The illustration shows half the plane through the symmetry axis. Vertical axis (z) corresponds to the longitudinal axis of the antenna; horizontal axis (r) corresponds to

In RFA for high temperature hyperthermia therapy in breast cancer several devices from different manufacturers were used in diverse ways for varying periods of time and assorted protocols, therefore exist heterogeneous results. Nevertheless successful cases for were obtained for smaller tumors with a low failures and complication rate. On the other hand the effect of MWA on malignant and normal adipose-dominated tissues of the breast was simulated using an axisymmetric electromagnetic model. This model can analyze the heating patterns using the bioheat equation. The results from computer modeling demonstrated that, effectively, the difference in dielectrical properties and thermal parameters between the malign and normal adipose-dominated tissue could cause the preferential heating on tumor during MWA. Even though electromagnetic high temperature hyperthermia requires further research, it is a promising minimally invasive modality for

The project described was supported by Instituto de Ciencia y Tecnología del Distrito Federal. Project Name: "Desarrollo de un sistema automatizado de determinación

radial direction.

**4. Conclusion** 

the local treatment of breast cancer.

**Acknowledgement** 

**Figure 3.** Temperature distribution of normal adipose-dominated breast tissue at a microwave power output of 10 W. The isotherm at 60 ºC is highlighted. The illustration shows half the plane through the symmetry axis. Vertical axis (z) corresponds to the longitudinal axis of the antenna; horizontal axis (r) corresponds to radial direction.

High Temperature Hyperthermia in Breast Cancer Treatment 97

**Figure 4.** Temperature distribution of breast cancer tissue at a microwave power output of 10 W. The isotherm at 60 ºC is highlighted. The illustration shows half the plane through the symmetry axis. Vertical axis (z) corresponds to the longitudinal axis of the antenna; horizontal axis (r) corresponds to radial direction.
