# Numerical Estimation of SAR and Temperature Distributions inside Differently Shaped Female Breast Tumors during Radio-Frequency Ablation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Female Breast Phantom Model

#### 2.2. Material Properties

^{3}. The electrodes were modeled as perfect electric conductor (PEC) materials.

#### 2.3. RF Applicator Model

_{0}= 25 V was assumed on the lower electrode (electrode 2), whereas the upper electrode (electrode 1) was grounded (V

_{0}= 0). The upper dielectric (dielectric 1) with a length of 58.5 mm and a diameter of 0.5 mm was surrounded by a plastic catheter measuring 57 mm in length and 0.7 mm in diameter, which served as a protective element.

#### 2.4. Equivalent Tumor Models

#### 2.5. Electro-Conductive Field and Generalized Laplace Equation

**J**and

**E**correspond to the vectors of current density (A/m

^{2}) and electric field strength (V/m), respectively, σ stands for the electric conductivity of the material (S/m), and φ means the electric potential (V).

_{0}/f ≈ 3 km) is much larger than the largest size of the analyzed RF applicator, and thus the displacement currents compared to the conduction currents are negligible [27]. Since the E-field pattern around the needle applicator is forced by the voltage applied to electrode 2 (see Figure 4b), the generalized Laplace equation in the following form can be used:

_{0}= 0) and electrode 2 was voltaged by electric potential φ = V

_{0}= 25 V (see Figure 4b); φ = V

_{0}= 0 was assigned to the external planes of the computational domain. The other boundaries, which result from the EM field theory and reflect the continuity of the normal components of the current density vector between two adjacent tissues, can be introduced as:

#### 2.6. Modified Pennes Bioheat Transfer Equation

^{3}) is the tissue mass density. The second term describes heat conduction in tissue with thermal conductivity k (W/m/K). The third term relates to the cooling effects of blood perfusion through the tissue expressed by the heat transfer rate HTR (mL/min/kg) as well as the difference between the current temperature of tissue T (K) and the arterial blood temperature T

_{b}(K) [61]. The next term describes heat losses induced by tissue metabolism. This element is proportional to the heat generation rate HGR (W/kg). The last term, often called external heat generation Q

_{ext}= ρSAR (W/m

^{3}), describes heat losses caused by the RF applicator. The SAR-based heat source measures the EM energy absorbed by the tissue unit mass during unit time. The SAR (W/kg) parameter is proportional to the tissue temperature [25,62], namely:

^{3}), mass m (kg), and density ρ (kg/m

^{3}), |

**E**| = |$\nabla $φ| (V/m) stands for the amplitude of electric field strength produced by the RF applicator voltaged by the electric potential φ (V), σ (S/m) is the electrical conductivity of the medium, and t (s) is the duration of the EM field exposure. The SAR parameter is a coupling of the modified Pennes bioheat equation (7) with the generalized Laplace equation (4) and plays an extremely important role in EM field dosimetry and human tissue safety [62,63].

_{b}(J/kg/K), blood density ρ

_{b}(kg/m

^{3}), tissue density ρ (kg/m

^{3}), and blood perfusion ω (1/s); as the temperature increases, the tumor perfusion decreases exponentially [57].

_{skin}(W/m/K) is the skin thermal conductivity of the breast phantom, T

_{ext}stands for the air temperature that surrounds the breast model, and

**n**relates to the normal vector perpendicular to the skin layer surface.

#### 2.7. SAR and Power Dissipation Values

^{3}) deposited in the target tissue region with complete mass M (kg) and volume V (m

^{3}). In general, two main approaches of SAR averages are commonly employed in numerical simulations, namely values averaged over some finite mass SAR

_{mass}(W/kg) or volume SAR

_{vol}(W/m

^{3}) as defined below:

**E**| = |$\nabla $φ| (V/m) stands for the amplitude of electric field strength produced by the RF applicator. These equations indicate that the averaged SAR values are scaled and related by the formula SAR

_{vol}= ρSAR

_{mass}. Knowing such values, it is possible to estimate power dissipation in the targeted tissue, including total power losses:

_{vol}value. Additionally, the peak spatial-average SAR (psSAR) for constant-mass cubes of tissue (e.g., 1 g) is defined according to the IEEE/IEC 62704-1 standard [65].

## 3. Results

_{max}), the SAR value averaged in unit mass (SAR

_{mass}) and volume (SAR

_{vol}) of various female breast tissues, including equivalent tumor models, as well as the SAR value averaged by 1 g mass of tissue (spSAR

_{1g}). Additionally, the total power losses induced inside individual female breast tissues are given according to the formulations in Section 2.7. The total mass and volume of the breast tissues and the values for the equivalent tumor models are included as well.

^{2}/K, T

_{ext}= 25 °C, and T

_{0}= T

_{b}= 37 °C [25,66,67], which correspond to the heat transfer coefficient, external temperature, and initial temperature, respectively. The induced steady-state isothermal surfaces for temperatures of 50 °C, 44 °C, and 38 °C (isosurface-50, isosurface-44, isosurface-38) obtained for the differently shaped tumors in the same RF applicator operating conditions (f = 100 kHz) are shown in Figure 7. Isosurace-50 is marked in red, isosurface-44 in pink, and isosurface-38 in blue. In this case, isosurface-50 was considered an ablation zone. Besides, the rendered equivalent tumor models were added.

## 4. Discussion

_{ISO-50}= 242.536 mm

^{3}) and in the ellipsoid E2 tumor model (V

_{ISO-50}= 201.253 mm

^{3}), respectively. However, the closest ablation value to the tumor model T (V

_{ISO-50}= 223.215 mm

^{3}) was achieved by the S1-equivalent tumor model (V

_{ISO-50}= 226.064 mm

^{3}). Considering the volume of the ablation zone of tumor T as a reference, it can be seen that the ablative zones did not change considerably (less than a 10% variation in volume). This result may suggest that the effectiveness of ablation does not depend on the utilized tumor model but mainly on the RF probe applied; therefore, equivalent tumor models can be used instead of irregularly shaped tumors.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Screening mammography of the breast tumor. Reproduced with kind permission from Dalian University of Technology, China.

**Figure 2.**Views through (

**a**) cross-section of the female breast phantom (z = 0) including the main tissues: breast fat (yellow), breast gland (blue), fat (green), muscle (orange), skin (red); and (

**b**) irregular breast tumor (pink) oriented in the x-y plane and (

**c**) oriented in the y-z plane.

**Figure 4.**Model of (

**a**) an RF applicator inserted into the irregular breast tumor, and (

**b**) the internal structure of the needle including two electrodes in different electric potential conditions, dielectric, and external catheter.

**Figure 5.**Analyzed scenarios of tumor shapes: (

**a**) irregular tumor T, equivalent sphere S

_{1}, and equivalent ellipsoid E

_{1}have the same surfaces and volumes; (

**b**) irregular tumor T is inscribed into sphere S

_{2}and ellipsoid E

_{2}with correspondingly larger volumes.

**Figure 6.**Normalized SAR distributions in the yz plane derived from the RF needle applicator for all analyzed equivalent tumor models: (

**a**) sphere S

_{1}; (

**b**) sphere S

_{2}; (

**c**) ellipsoid E

_{1}; (

**d**) ellipsoid E

_{2}; (

**e**) irregular tumor; (

**f**) case without tumor.

**Figure 7.**Isosurfaces of temperature in the yz plane derived from the RF needle applicator for all analyzed breast tumor equivalent models: (

**a**) sphere S

_{1}; (

**b**) sphere S

_{2}; (

**c**) ellipsoid E

_{1}; (

**d**) ellipsoid E

_{2}; (

**e**) irregular tumor; (

**f**) case without tumor.

**Figure 8.**Temperature distributions along different axes of analyzed equivalent tumor models: (

**a**) x-axis; (

**b**) y-axis; (

**c**) z-axis, where S

_{1}and S

_{2}are spherically equivalent tumor models; and E

_{1}and E

_{2}ellipsoidal equivalent tumor models.

**Figure 9.**Time-dependent temperature distributions in the center of various tumors, including zoom view inside, where S

_{1}and S

_{2}correspond to spherical tumor models and E

_{1}and E

_{2}correspond to ellipsoidal tumor models.

**Figure 10.**Grid element number, total degree of freedoms, peak memory usage, and clock time values for an electro-quasi-static solver (

**a**,

**c**); and a thermal solver (

**b**,

**d**) in the case of all analyzed tumor models.

**Table 1.**Electro-thermal female breast tissue parameters valid for RF ablation treatment with frequency f = 100 kHz [60].

Tissue | σ (S/m) | ρ (kg/m ^{3}) | C (J/kg/K) | k (W/m/K) | HTR * (mL/min/kg) | HTR * (W/m ^{3}) | HGR ** (W/kg) |
---|---|---|---|---|---|---|---|

Blood | 0.7030 | 1050 | 3617 | 0.517 | 10,000 | 6.646 × 10^{5} | 0 |

Breast fat | 0.0250 | 911 | 2348 | 0.209 | 47 | 2710 | 0.728 |

Breast gland | 0.5370 | 1041 | 2960 | 0.334 | 150 | 9884 | 2.323 |

Fat | 0.0434 | 911 | 2348 | 0.211 | 33 | 1903 | 0.507 |

Muscle | 0.3618 | 1090 | 3421 | 0.495 | 37 | 2553 | 0.906 |

Skin | 0.0005 | 1109 | 3391 | 0.372 | 106 | 7441 | 1.648 |

Tumor | 0.3618 | 1090 | 3437 | 0.563 | Equation (9) | Equation (9) | 12 |

Breast Tumor Tissues | x-axis Length a (mm) | y-axis Length b (mm) | z-axis Length c (mm) | Total Surface Area A _{calc} (cm^{2}) | Total Surface Area A _{meas} (cm^{2}) | Total Volume V _{calc} (cm^{3}) | Total Volume V _{meas} (cm^{3}) | Total Mass m _{calc} (g) | Total Mass m _{meas} (g) |
---|---|---|---|---|---|---|---|---|---|

Ellipsoid E_{1} | 16.412 | 29.354 | 10.866 | 13.153 | 12.716 * | 2.740 | 2.749 * | 2.989 | 2.998 * |

Sphere S_{1} | 17.474 | 17.474 | 17.474 | 9.593 | 9.538 * | 2.794 | 2.771 * | 3.046 | 3.021 * |

Ellipsoid E_{2} | 26.574 | 42.588 | 27.675 | 31.498 | 32.153 * | 16.395 | 16.310 * | 17.882 | 17.790 * |

Sphere S_{2} | 36.000 | 36.000 | 36.000 | 40.715 | 40.484 * | 24.429 | 24.270 * | 26.637 | 26.460 * |

Tumor T | 38.630 | 24.269 | 24.269 | 12.320 ** | 12.722 * | 2.844 ** | 2.791 * | 3.101 ** | 3.043 * |

**Table 3.**Calculated power losses inside breast tissues during RF ablation treatment with frequency f = 100 kHz.

Breast Tissues | Total Mass m (g) | Total Volume V (cm ^{3}) | Maximum Local SAR SAR _{max} (W/kg) | Mass-Average Local SAR SAR _{mass} (W/kg) | Total Loss Power Density p = SAR _{vol} (W/m^{3}) | Spatial-Average Local SAR ** psSAR _{1g} (W/kg) | Total Loss Power P (W) |
---|---|---|---|---|---|---|---|

Breast fat * | 4766.2 | 5232.1 | 9560 | 4.95 × 10^{−3} | 13.525 | 650.82 | 0.0254 |

Breast gland * | 814.8 | 783.0 | 2,7440 | 7.099 | 7388.65 | 699.33 | 1.1485 |

Fat * | 441.3 | 484.4 | 11,430 | 4.41 × 10^{−2} | 40.153 | 1050.0 | 0.0195 |

Muscle * | 731.3 | 670.6 | 1.61 × 10^{−6} | 8.24 × 10^{−7} | 8.99 × 10^{−4} | 1.58 × 10^{−6} | 6.0 × 10^{−7} |

Skin * | 495.6 | 446.9 | 4.10 × 10^{−4} | 8.31 × 10^{−7} | 9.22 × 10^{−4} | 0.142 | 19.524 |

Ellipsoid E_{1} | 2.998 | 2.749 | 152,700 | 326.435 | 355 984 | 823.41 | 0.9786 |

Sphere S_{1} | 3.021 | 2.771 | 151,900 | 327.576 | 357,163 | 825.05 | 0.9897 |

Ellipsoid E_{2} | 17.79 | 16.31 | 155,600 | 58.844 | 64,175 | 833.98 | 1.0467 |

Sphere S_{2} | 26.46 | 24.27 | 155,800 | 39.733 | 43,325 | 835.40 | 1.0515 |

Tumor T | 3.043 | 2.791 | 167,200 | 327.894 | 357,506 | 824.83 | 0.9978 |

Breast Tissues | Ablation Volume V _{ISO-50} (mm^{3}) | Relative Error δ (%) |
---|---|---|

Ellipsoid E_{1} | 242.536 | 8.656 |

Sphere S_{1} | 226.064 | 1.276 |

Ellipsoid E_{2} | 201.253 | 9.839 |

Sphere S_{2} | 204.365 | 8.445 |

Tumor T | 223.215 | – |

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**MDPI and ACS Style**

Miaskowski, A.; Gas, P. Numerical Estimation of SAR and Temperature Distributions inside Differently Shaped Female Breast Tumors during Radio-Frequency Ablation. *Materials* **2023**, *16*, 223.
https://doi.org/10.3390/ma16010223

**AMA Style**

Miaskowski A, Gas P. Numerical Estimation of SAR and Temperature Distributions inside Differently Shaped Female Breast Tumors during Radio-Frequency Ablation. *Materials*. 2023; 16(1):223.
https://doi.org/10.3390/ma16010223

**Chicago/Turabian Style**

Miaskowski, Arkadiusz, and Piotr Gas. 2023. "Numerical Estimation of SAR and Temperature Distributions inside Differently Shaped Female Breast Tumors during Radio-Frequency Ablation" *Materials* 16, no. 1: 223.
https://doi.org/10.3390/ma16010223