# Numerical Investigation of a Thermal Ablation Porous Media-Based Model for Tumoral Tissue with Variable Porosity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{1}= 3.10 cm defines the tissue boundary, while the inner sphere radius r

_{2}= 0.620 cm is the heating volume; hence, the total heating volume is equal to 1 cm

^{3}.

_{var}is the variable porosity in the tumoral tissue, namely, the blood volume fraction in the entire tissue volume, and it will be described in the following paragraph, ρ is the density, c the specific heat, T

_{t}and T

_{b}are the tumoral tissue and blood temperatures, respectively, t is the time,

**u**is the blood velocity vector, h

_{c}is the interfacial heat transfer coefficient, and a is the volumetric transfer area between tissue and blood, evaluated from the hydraulic diameter definition in porous media as

_{min}= 0.07 and ε

_{max}= 0.23. In Figure 3 the variable porosity field in a slice of the spherical domain is displayed. From the figure it is clear the increase of porosity from the very small values in the core to the larger values in the rim, highlighting the different vascularization between the two zones. In the next Results Section, the variable porosity case will be compared to uniform porosities set equal to 0.07 and 0.23, and therefore, the minimum and maximum values of the experimental measures range.

_{c}, it is assumed to be constant and equal to 170 W m

^{−2}K

^{−1}, as in Yuan [26]. Table 1 summarizes all the blood velocities with the related investigated blood vessel diameters.

_{ext}in Equations (1) and (2) represents a generic external power density applied to the tumoral tissue during the treatment, and the symbol <> means that the volume-averaged quantity of a generic variable is considered. Concerning the fluid phase, a uniform blood velocity is assumed in all directions in order to reproduce a more realistic in vivo vessel network.

^{43}s

^{−1}and ΔE = 2814 × 10

^{5}J mol

^{−1}, as in [28]. These parameters are available for different tissues, and they have been obtained fitting known exposure times and temperatures with cell surviving probabilities. In the Arrhenius model, if Ω = 1, a 68% of cell death probability is achieved; hence, to have a more accurate prediction of the coagulation zone dimensions, thermal damage is evaluated using the D99 thermal damage contour, considering the isoline at Ω = 4.6, which fits for obtaining the 99% cell death probability. Thus, the β coefficient will be 1 for Ω < 4.6 and 0 for Ω = 4.6.

_{fg}is the product of water latent heat of vaporization and water density at 100 °C (2.17 × 10

^{9}J m

^{−3}), and C

_{w}is the water content inside the tumoral liver tissue (81%) or in the blood (79%), as found in the literature [30,31]. The temperature difference ΔT

_{b,t}is assumed to be 1 °C [29].

_{ext}= 5 × 10

^{6}W m

^{−3}with a heating time of 100 s, and hence, the resulting total energy provided to the tissue is 500 J.

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Volumetric transfer area (m^{−1}) |

A | Frequency factor (s^{−1}) |

c | Specific heat (J kg^{−1} K^{−1}) |

C_{w} | Water content (%) |

d | Diameter of the blood vessel (m) |

d_{c} | Coagulation diameter (m) |

h_{c} | Interfacial heat transfer coefficient(W m^{−2} K^{−1}) |

h_{fg} | Product of water latent heat of vaporization and water density at 100 °C(J m^{−3}) |

k | Thermal conductivity (W m^{−1} K^{−1}) |

Q_{ext} | External power density (W m^{−3}) |

r | Radial coordinate (m) |

r_{1} | Radius of the external sphere (m) |

r_{2} | Radius of the internally heated sphere (m) |

R | Universal gas constant (J mol^{−1} K^{−1}) |

t | Time (s) |

T | Temperature (K) |

u | Blood velocity vector (m s^{−1}) |

x, y, z | Spatial coordinates (m) |

Greek symbols | |

β | Coefficient (-) |

ΔE | Activation energy (J mol^{−1}) |

ε | Porosity (-) |

ρ | Density (kg m^{−3}) |

Ω(t) | Degree of tissue death (-) |

Subscripts | |

b | Blood |

max | Maximum |

min | Minimum |

t | Tissue |

tot | Total |

var | Variable |

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**Figure 3.**Variable porosity field in tumoral tissue domain (

**a**) and porosity profile on the sphere radius (

**b**).

**Figure 4.**Tissue temperatures profiles evaluated at the center of the sphere for current work and [26]: (

**a**) ε = 0.005, Q

_{ext}= 2 × 10

^{6}W m

^{−3}and (

**b**) ε = 0.05, Q

_{ext}= 50 × 10

^{6}W m

^{−3}.

**Figure 5.**Axisymmetric view of thermal damage percentage evaluated after 100 s of thermal treatment for the three different porosity models and the four different blood vessels: (

**a**) capillaries, (

**b**) terminal arteries, (

**c**) terminal branches, and (

**d**) tertiary branches.

**Figure 6.**Maximum tissue temperature values reached considering the three different porosities and the four blood vessels investigated.

**Figure 7.**Tissue temperature fields at the end of heating time for uniform ε = 0.07 (

**a**–

**d**), uniform ε = 0.23 (

**e**–

**h**), and εvar (

**i**–

**l**). Capillaries (d = 8 μm), terminal arteries (d = 30 μm), terminal branches (d = 50 μm), and tertiary branches (d = 140 μm) are taken into account. Coagulation contours delimited by the black lines.

d (μm) | u (cm s^{−1}) | |
---|---|---|

Capillaries | 8 | 0.07 |

Terminal arteries | 30 | 0.40 |

Terminal branches | 50 | 2.00 |

Tertiary branches | 140 | 3.40 |

ρ (kg·m^{−3}) | c (J·kg^{−}^{1}·K^{−}^{1}) | k (W·m^{−}^{1}·K^{−}^{1}) | |
---|---|---|---|

Tumor | 1045 | 3760 | 0.600 |

Blood | 1000 | 3639 | 0.502 |

Number of Triangular Elements | Maximum Tissue Temperature |
---|---|

2192 | 99.101 °C |

4384 | 99.100 °C |

8768 | 99.098 °C |

17536 | 99.097 °C |

Time Step | Maximum Tissue Temperature |
---|---|

0.05 s | 99.094 °C |

0.10 s | 99.098 °C |

0.20 s | 99.184 °C |

0.40 s | 99.187 °C |

Coagulation Diameter d_{c} (cm) | |||
---|---|---|---|

ε = 0.07 | ε = 0.23 | ε_{var} | |

Capillaries | 1.40 | 1.40 | 1.40 |

Terminal arteries | 1.40 | 1.32 | 1.40 |

Terminal branches | 1.24 | 0.81 | 1.20 |

Tertiary branches | 1.20 | 0.00 | 1.16 |

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**MDPI and ACS Style**

Andreozzi, A.; Brunese, L.; Iasiello, M.; Tucci, C.; Vanoli, G.P.
Numerical Investigation of a Thermal Ablation Porous Media-Based Model for Tumoral Tissue with Variable Porosity. *Computation* **2021**, *9*, 50.
https://doi.org/10.3390/computation9050050

**AMA Style**

Andreozzi A, Brunese L, Iasiello M, Tucci C, Vanoli GP.
Numerical Investigation of a Thermal Ablation Porous Media-Based Model for Tumoral Tissue with Variable Porosity. *Computation*. 2021; 9(5):50.
https://doi.org/10.3390/computation9050050

**Chicago/Turabian Style**

Andreozzi, Assunta, Luca Brunese, Marcello Iasiello, Claudio Tucci, and Giuseppe Peter Vanoli.
2021. "Numerical Investigation of a Thermal Ablation Porous Media-Based Model for Tumoral Tissue with Variable Porosity" *Computation* 9, no. 5: 50.
https://doi.org/10.3390/computation9050050