# Three-Phase-Lag Bio-Heat Transfer Model of Cardiac Ablation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{3}), c is the specific heat capacity (J/kg/K), T is the tissue temperature (K), k is the thermal conductivity (W/m/K), ρ

_{b}is the density of blood (kg/m

^{3}), c

_{b}is the specific heat capacity of blood (J/kg/K), ω

_{b}is the blood perfusion rate (1/s), T

_{b}is the temperature of blood entering the tissue, the term [ρ

_{b}c

_{b}ω

_{b}(T − T

_{b})] represents the heat sink effect caused by the small capillary vasculature, Q

_{m}is the metabolic heat generation (W/m

^{3}), which is normally ignored due to its minimal impact compared to other heat source terms during thermal ablation, and Q

_{p}(W/m

^{3}) represents the volumetric heat generation during thermal ablative procedures (e.g., radiofrequency or laser heat sources).

_{q}is the thermal relaxation time that represents the time delay between the heat flux vector and the temperature gradient. The constitutive relation obtained by incorporating the thermal relaxation time phase lag as the linear extension of Fourier law is known as the single-phase-lag (SPL) non-Fourier heat transfer model (or Cattaneo-Vernotte (C-V) heat transfer model). In this case, the resulting hyperbolic-type SPL heat equation is given by:

_{q}is the phase lag for the heat flux, and τ

_{t}is the phase lag for the temperature gradient. For τ

_{q}< τ

_{t}, heat flux precedes the temperature gradient, and for τ

_{q}> τ

_{t}, the temperature gradient precedes the heat flux. It is noteworthy to mention that in order to obey the causality principle τ

_{q}≥ τ

_{t}, as highlighted in [13]. By substituting Equation (2) in Equation (5), and after some mathematical manipulations, the dual-phase-lag (DPL) non-Fourier heat transfer model can be obtained as:

_{t}= 0, Equation (6) reduces to the SPL bio-heat transfer model (Equation (4)), and further reduces to Fourier’s heat transfer model (Equation (2)) for τ

_{q}= 0.

_{q}, τ

_{t}, and τ

_{v}are phase lags associated with heat flux, temperature gradient, and thermal displacement gradient, respectively, v is thermal displacement satisfying $\dot{v}=T$, and ${k}^{*}$ is the rate of thermal conductivity. After some mathematical manipulations of Equation (7) combined with Equation (2), the following constitutive equation for TPL can be obtained [33]:

_{v}< τ

_{t}< τ

_{q}[38]. For k* = 0 and integrating w.r.t time, Equation (8) will reduce to the DPL bio-heat transfer model. Further with τ

_{t}= 0, this equation will reduce to the SPL model, and additionally, with τ

_{q}= 0, the equation will reduce to the Pennes’ bio-heat transfer model.

_{p}) would be zero in the TPL model (Equation (8)). To avoid this paradox, we have considered a constant temperature at the tip of the applicator, which is maintained at 95 °C. It is noteworthy to mention that the constant temperature can be easily maintained during the thermal ablative procedures utilizing a closed-loop feedback proportional-integral-derivative (PID) controller, as demonstrated in previous studies [39,40,41,42,43]. A comparative analysis has been conducted by comparing the temperature distributions predicted within the myocardium tissue incorporating Pennes’ (Equation (2)), SPL (Equation (4)), DPL (Equation (6)), and TPL (Equation (8)) bio-heat transfer models.

_{b}is the density of blood (kg/m

^{3}), p is the blood pressure (Pa), µ is the dynamic viscosity of blood (=2.1 × 10

^{−3}Pa.s), and F is the body force neglected in this study [43,44,45]. It is noteworthy to mention that Newtonian behavior has been used to model the blood, which is adequate for shear rates >100 s

^{−1}[46,47]. The fluid–structure interaction (FSI) has received significant attention in research related to cardiovascular modeling [48,49,50,51,52,53,54,55]. In this study, FSI only pertains to the convective heat-sink (cooling) effect induced by the blood flow during cardiac ablation, as the cardiac tissue has been modeled as a rigid body. However, future studies are warranted to incorporate the mechanical deformation of the cardiac tissue caused by the pulsatile blood flow within the cardiac chamber, as has been recently reported in [56,57].

^{3}, c = 3111 J/kg/K, ρ

_{b}= 1000 kg/m

^{3}, c

_{b}= 4180 J/kg/K, ω

_{b}= 0.017 s

^{−1}, k = 0.54 W/m/K, k* = 0.1 W/m/K/s, τ

_{q}= 16 s, τ

_{T}= 6 s and τ

_{v}= 2 s [19,33,43,44,45,58]. The computational domain presented in Figure 1b was discretized using the heterogeneous tetrahedral mesh elements with additional refinements close to the applicator where higher thermal gradients were expected. Furthermore, a grid independence study was performed to determine the optimal number of mesh elements that would result in a mesh-independent solution. The mesh refinements were carried out using the ablation volume convergence criterion, i.e., the mesh was progressively refined until the absolute error for ablation volume was less than 0.5 % compared to the previous mesh size. The final mesh comprises of 551,690 domain elements, 26,674 boundary elements, and 1288 edge elements. The constitutive equations for heat transfer and fluid flow, presented earlier, were solved using the multifrontal massively parallel sparse direct solver (MUMPS) [61], employing a segregated solution approach. All of the simulations were conducted on a Dell Precision 7920 Tower workstation with 96 GB RAM and 2.20 GHz Intel

^{®}Xeon

^{®}processors with a computational time of less than 4 h for each simulation.

## 3. Results and Discussion

^{3}, specific heat capacity of tissue = 3600 J/kg/K, density of the blood = 1060 kg/m

^{3}, specific heat capacity of the blood = 3770 J/kg/K, the blood perfusion rate = 1.87 × 10

^{−3}s

^{−1}, thermal conductivity of the tissue = 0.235 W/m/K, the rate of thermal conductivity = 0.1 W/m/K/s, phase lag associated with heat flux = 16 s, phase lag associated with temperature gradient = 6 s, and the phase lag associated with thermal displacement gradient = 2 s. The comparative analysis of the present results with that of [33] is presented in Figure 2. As evident from Figure 2, a good agreement has been obtained between the results predicted from the present model with those reported in [33]. Hence the TPL model used in the present study is efficient and lends great confidence to the results reported in the next sections.

_{q}), dual-phase-lag (τ

_{t}), and three-phase-lag (τ

_{v}) on the thermal behavior obtained with the TPL model is presented in Figure 8 and Figure 9. The effect of τ

_{q}on temperature response at a point 1 mm below the tip of the applicator is presented in Figure 8a. As evident from this figure, the higher the value of τ

_{q}, the higher would be the lagging characteristic of thermal behavior that would lead to a steeper rise in temperature. Thus, the heat propagation within the tissue decreases with an increase in the value of τ

_{q}.

_{t}on the temperature response at a point 1 mm below the tip of the applicator is presented in Figure 8b. As evident from Figure 8b, the influence of variation in τ

_{t}is smaller on the thermal propagation and lag characteristics as compared to τ

_{q}. Figure 8c presents the influence of variation in the thermal displacement gradient τ

_{v}on the temperature response at a point 1 mm below the tip of the applicator. As evident from Figure 8c, the increase in the value of τ

_{v}results in a decrease in the predicted temperature during cardiac ablation. The effect of the influence of variation in τ

_{q}, τ

_{t}, and τ

_{v}on the ablation volumes predicted after 10 s, 30 s, and 60 s of cardiac ablation are presented in Figure 9. As evident from this figure, the maximum variation in the ablation volume for different values of lags occurs at 10 s and then decreases with the increase in treatment time.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Reduction in the computational domain from the full thorax model to a limited domain model (Adapted from Irastorza et al. [37] under the terms of the Creative Commons CC BY license for an open access article). (

**b**) Schematic of the three-dimensional model of cardiac ablation that comprises of myocardium tissue, blood chamber, and the applicator.

**Figure 2.**Comparison of the predicted results of the present TPL model with that of Zhang et al. (2020) [33]: (

**a**) temporal evolution of temperature at a location of 4.5 mm; and (

**b**) temperature distribution at time t = 50 s.

**Figure 3.**Temporal evolution of temperature predicted with different bio-heat transfer models at a location of: (

**a**) 0.5 mm, (

**b**) 1 mm, and (

**c**) 2 mm below the tip of the applicator.

**Figure 4.**Temperature distribution predicted with different bio-heat transfer models along the line below the tip of the applicator at: (

**a**) 30 s, and (

**b**) 60 s.

**Figure 5.**Comparison of the ablation volume predicted with different bio-heat transfer models at 10 s, 30 s, and 60 s of treatment times.

**Figure 6.**(

**a**) Temporal evolution of temperature predicted with and without considering the blood perfusion effects at a location of 1 mm below the applicator’s tip. Temperature distribution predicted with and without considering the blood perfusion effects along the line below the tip of the applicator at: (

**b**) 30 s and (

**c**) 60 s.

**Figure 7.**Temporal variation of ablation volume attained within the myocardium tissue with low and high blood flow conditions within the cardiac chamber.

**Figure 8.**Effect of magnitude of (

**a**) τ

_{q}, (

**b**) τ

_{t}, and (

**c**) τ

_{v}on the time-history of temperature at a location of 1 mm below the tip of the applicator.

**Figure 9.**Comparison of the ablation volume attained after 10 s, 30 s, and 60 s of cardiac ablation for different values of (

**a**) τ

_{q}, (

**b**) τ

_{t}and (

**c**) τ

_{v}.

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**MDPI and ACS Style**

Singh, S.; Saccomandi, P.; Melnik, R.
Three-Phase-Lag Bio-Heat Transfer Model of Cardiac Ablation. *Fluids* **2022**, *7*, 180.
https://doi.org/10.3390/fluids7050180

**AMA Style**

Singh S, Saccomandi P, Melnik R.
Three-Phase-Lag Bio-Heat Transfer Model of Cardiac Ablation. *Fluids*. 2022; 7(5):180.
https://doi.org/10.3390/fluids7050180

**Chicago/Turabian Style**

Singh, Sundeep, Paola Saccomandi, and Roderick Melnik.
2022. "Three-Phase-Lag Bio-Heat Transfer Model of Cardiac Ablation" *Fluids* 7, no. 5: 180.
https://doi.org/10.3390/fluids7050180