# Fluid–Structure Interaction and Non-Fourier Effects in Coupled Electro-Thermo-Mechanical Models for Cardiac Ablation

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

_{q}and τ

_{t}) can be represented as [4,24].

_{q}is the thermal relaxation time that represents the time delay between heat flux and the associated heat conduction through a medium (i.e., the delay arising from thermal inertia) [21,25], τ

_{t}is the phase lag in establishing temperature gradient across the medium (i.e., the delay arising from micro-structural interactions). While the models accounting for thermal relaxation in the context of hyperbolic thermo-elasticity have a longer history [26,27,28], a comprehensive analysis and applications of such models in the context of thermal ablation of biological tissues are fairly new [21]. Due to two associated phase lags, Equation (1) is often referred to as a dual phase lag (DPL) model that will be reduced to single phase lag (SPL) for τ

_{q}≠ 0, τ

_{t}= 0, and Fourier’s law based Pennes bioheat transfer equation for τ

_{q}= τ

_{t}= 0. ρ is the density, c is the specific heat capacity, k is the thermal conductivity, $\overrightarrow{\mathrm{u}}$ is the velocity vector. T is the temperature to be computed during RF-assisted cardiac ablation. Q

_{p}is the resistive heat generated due to RF heating and under quasi-static approximation of Maxwell’s equation, it is represented as the product of current density (J) and electric filed intensity (E), i.e., Q

_{p}= J$\cdot $E = σE

^{2}(W/m

^{3}), where σ is the electrical conductivity of the biological tissue. The effective heat capacity and the thermal conductivity of porous cardiac tissue are given by:

_{x}, u

_{y}, u

_{z}) is the velocity vector, ρ

_{b}is the density of blood, P is the blood pressure, µ is the dynamic viscosity of blood (= 2.1 × 10

^{−3}Pa·s), ε is the porosity and K is the permeability of porous cardiac tissue (= 7.7 × 10

^{−11}m

^{2}). Other models for the fluid part of the fluid–structure interaction (FSI) problem can be easily incorporated in the proposed framework, including those based on non-Newtonian flows (e.g., [30,31,32,33,34]). Although not yet fully integrated with the models for ablation therapies, the cardiac fluid dynamics models have been an area of active research from theoretical, numerical, and experimental perspectives [35,36,37]. Likewise, FSI problems have received significant attention of the research community [38,39,40,41,42,43,44,45,46]. In our present context, it should be noted that while the research in cardiovascular modeling also covers cardiac electro-mechanical coupling, specifics of ablation problems require new models and new approaches for their solution.

^{T}+ ∇$\overline{u}$)/2] is the strain tensor, $\overline{u}$ is the mechanical displacement vector, ε

^{th}= $\left({\displaystyle \underset{{T}_{ref}}{\overset{T}{\int}}\alpha dT}\right)$ is the thermal strain, α is the thermal expansion coefficient, T is the temperature computed from bio-heat transfer model, T

_{ref}= 37 °C is the reference temperature and δ is the Kronecker delta function given by:

^{®}Xeon

^{®}E5-2680 v2 @ 2.80 GHz processor.

## 3. Results and Discussion

#### 3.1. Coupled Electro-Thermo-Mechanical Model of Cardiac Ablation

_{in}= 3 cm/s, thermal lags of τ

_{q}= 8 s, τ

_{t}= 0.045 s, insertion depth of 2 mm and perpendicular orientation of electrode. As evident from Figure 2a–c, the temperature distribution within the cardiac tissue is of the ellipsoid shape which is consistent with the other modeling studies available in the literature. Further, due to the blood flow in the cardiac chamber (from left to right face), the temperature distribution is not symmetric along the electrode insertion axis. Rather, the temperature distribution in the downstream of the electrode is slightly higher as compared to the upstream side. This can be attributed to the fact that at the upstream side the RF electrode obstructs the path of the blood flow within the blood chamber and accordingly the blood velocity will be slightly higher at upstream side as compared to downstream side of the electrode. This could eventually lead to an enhanced heat-sink effect whereby the heat generated due to the high-frequency RF currents is being carried away by the flowing blood.

#### 3.2. Effect of Porosity

#### 3.3. Effect of Thermal Relaxation Times

_{q}and τ

_{t}. Importantly, τ

_{q}refers to the phase lag due to heat flux, i.e., time delay between the heat flux and the temperature variation. By virtue of this lag, the thermal energy propagation within the cardiac tissue during RF ablation is arrested. A higher value of τ

_{q}results in higher energy accumulation within the biological tissue and consequently significantly higher vibration characteristics in response to the elevated temperatures; while τ

_{t}refers to the phase lag due to the temperature gradient, i.e., heat flux vector precedes the temperature gradient. This lag eventually results in lower energy accumulation and lower peak temperatures within the biological tissue, and accordingly the increase in τ

_{t}results in diminishing vibration characteristic in thermal response [53,54]. Motivated by [21,53,54], three values of thermal lag have been selected for the biological tissue, viz., τ

_{q}= τ

_{t}= 0 (Fourier model), 1, 5. The effect of thermal relaxation times (τ

_{q}and τ

_{t}) on the temperature distribution and ablation volume for cardiac ablation procedure has been presented in Figure 4 for ε = 0.3 and u

_{in}= 3 cm/s. Figure 4a presents the temporal variation of temperature at a point 1 mm away from the RF electrode tip along the insertion axis for different values of thermal relaxation times.

_{q}and τ

_{t}. Eventually, beyond 30 s of ablation, the deviation between the predicted temperature for different values of thermal relaxation time diminishes with the passage of time. Similar variations have been seen in the ablation volume profile presented in Figure 4d. Moreover, the non-Fourier characteristics have a negligible effect on the maximum tissue and blood temperatures during the 60 s long cardiac ablation, as depicted in Figure 4b,c, respectively. Thus, for shorter duration of cardiac ablation, less than 30 s or so, the thermal relaxation times could significantly affect the predicted ablation volume, while for treatment times greater than 30 s, ablation volume is not sensitive to the value of thermal relaxation times.

#### 3.4. Effect of Blood Flow in the FSI Problem and Electrode Insertion Depth

_{in}= 3 cm/s) and high blood flow rate (u

_{in}= 8.5 cm/s). As evident from Figure 5a,b, the temperature distribution and ablation volume profiles for low blood flow rate conditions are always on a higher side when compared to the high blood flow rate conditions, except for first 10 to 15 s of cardiac ablation, whereby the deviations among low and high blood flow rate conditions are quite negligible.

#### 3.5. Effect of of Electrode Orientation

_{in}= 3 cm/s. As evident from this figure, the maximal depth of the lesion obtained after 60 s of cardiac ablation is nearly same for all orientations, while the maximal width of the lesion increases by decreasing the RF electrode orientation w.r.t tissue from 90° to 0°. This can be attributed to the fact that as the electrode orientation is decreased the contact surface area between the electrode tip and the cardiac tissue increases leading to enhanced delivery and accumulation of the RF energy within the tissue, thereby increasing the ablation zone attained after 60 s of the RF assisted cardiac ablation. The propagation of ablation volume after 15 s, 30 s and 60 s of cardiac ablation procedure has been presented in Figure 7 for different orientations of the electrode considered in the present study. Furthermore, the effects of electrode orientation on the maximum temperature distribution attained within the tissue and blood during 60 s of cardiac ablation have been presented in Figure 8a,b, respectively. As evident from these figures, the effect of electrode orientation is far more pronounced on the maximal blood temperature as compared to the tissue temperature. The maximal temperature profile is higher for the oblique orientation and lower for the parallel orientation of the RF electrode. Moreover, the temporal variations of the ablation volume for different orientations of the electrode have been presented in Figure 8c,d for low and high blood flow rates, respectively. The ablation volumes attained after 60 s of the RF assisted cardiac ablation for oblique and parallel orientations have been found to be 16.59% and 54.72% higher for low blood flow condition, respectively, and 17.92% and 63.73% for high blood flow condition, respectively, when compared to the perpendicular orientation.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Reduction of computational domain from full thorax model to limited domain model (reproduced from Irastorza et al. [17] under the terms of the Creative Commons CC BY license for an open access article). (

**b**) Boundary conditions associated with the limited-domain model considered in the present numerical study. (

**c**) Schematic of three-dimensional model derived from the selected control volume of limited-domain model.

**Figure 2.**Temporal evolution of: (

**a**–

**c**) temperature, (

**d**–

**f**) von Mises stress, and (

**g**–

**i**) volumetric strain distributions during cardiac ablation procedure.

**Figure 3.**Temporal variation of: (

**a**) temperature at a point 1 mm away from tip of electrode, (

**b**) maximum tissue temperature, (

**c**) maximum blood temperature, and (

**d**) ablation volume for different values of porosity.

**Figure 4.**Temporal variation of: (

**a**) temperature at a point 1 mm away from tip of electrode, (

**b**) maximum tissue temperature, (

**c**) maximum blood temperature, and (

**d**) ablation volume for different values of thermal relaxation times.

**Figure 5.**Temporal variation of: (

**a**) temperature at a point 1 mm away from tip of electrode, and (

**b**) ablation volume for different values of blood flow rates. (

**c**) Comparison of ablation volumes predicted with different insertion depth of the electrodes for low blood flow rate and (

**d**) high blood flow rate conditions.

**Figure 6.**(

**a**) Schematic of three different orientations of the RF electrode with respect to cardiac tissue. Temperature distribution obtained after 60 s of cardiac ablation with: (

**b**) perpendicular (90°), (

**c**) oblique (45°), and (

**d**) parallel (0°) orientations of the electrode.

**Figure 7.**Propagation of ablation zone during 60 s of cardiac ablative procedure for: (

**a**–

**c**) perpendicular (90°), (

**d**–

**f**) oblique (45°), and (

**g**–

**i**) parallel (0°) configuration of electrode.

**Figure 8.**Temporal variation of: (

**a**) maximum tissue temperature, and (

**b**) maximum blood temperature for different orientations of electrodes. Comparison of ablation volumes predicted with different orientations of the electrodes for: (

**c**) low blood flow rate, and (

**d**) high blood flow rate conditions.

Parameter | Myocardium/ Cardiac Tissue | Blood/ Cardiac Chamber | Electrode (Active Part) | Catheter (Insulated Part) |
---|---|---|---|---|

Density ρ (kg/m ^{3}) | 1060 | 1000 | 21,500 | 70 |

Specific heat c (J/(kg·K)) | 3111 | 4180 | 132 | 1045 |

Thermal conductivity k (W/(m·K)) | 0.53 | 0.541 | 71 | 0.026 |

Electrical conductivity σ (S/m) | 0.54 | 0.667 | 4.6 × 10^{6} | 10^{−5} |

Thermal expansion coefficient α (K ^{−1}) | 1 × 10^{−4} | – | – | – |

**Table 2.**Validation of the predicted results of current model with that of [6].

Parameter | Low Blood Flow u = 3 cm/s | High Blood Flow u = 8.5 cm/s | ||
---|---|---|---|---|

Previous Study [6] | Present FEM Study | Previous Study [6] | Present FEM Study | |

Depth of lesion | 7.64 mm | 7.62 mm | 7.53 mm | 7.53 mm |

Maximum width of lesion | 12.81 mm | 12.80 mm | 11.92 mm | 11.90 mm |

Maximum temperature in the tissue | 111.3 °C | 110.8 °C | 102.4 °C | 101.9 °C |

Maximum temperature in the blood | 89.6 °C | 89.2 °C | 79.9 °C | 79.7 °C |

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**MDPI and ACS Style**

Singh, S.; Melnik, R.
Fluid–Structure Interaction and Non-Fourier Effects in Coupled Electro-Thermo-Mechanical Models for Cardiac Ablation. *Fluids* **2021**, *6*, 294.
https://doi.org/10.3390/fluids6080294

**AMA Style**

Singh S, Melnik R.
Fluid–Structure Interaction and Non-Fourier Effects in Coupled Electro-Thermo-Mechanical Models for Cardiac Ablation. *Fluids*. 2021; 6(8):294.
https://doi.org/10.3390/fluids6080294

**Chicago/Turabian Style**

Singh, Sundeep, and Roderick Melnik.
2021. "Fluid–Structure Interaction and Non-Fourier Effects in Coupled Electro-Thermo-Mechanical Models for Cardiac Ablation" *Fluids* 6, no. 8: 294.
https://doi.org/10.3390/fluids6080294