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Mathematical study of a new coupled electro-thermo radiofrequency model of cardiac tissue

Mostafa Bendahmane, Youssef Ouakrim, Yassine Ouzrour, Mohamed Zagour

TL;DR

The paper develops a coupled electro-thermo-fluid model for radiofrequency ablation in cardiac tissue, partitioning the domain into blood and tissue regions and enforcing interface conditions that couple Navier–Stokes with heat and potential equations. It proves the global existence of weak solutions in three dimensions using a Faedo–Galerkin approximation, uniform energy estimates, and compactness arguments, while addressing the nonlinear, temperature-dependent conductivities and electrode boundary conditions. Numerical validation is provided via two-dimensional finite element simulations with domain decomposition and decoupled time stepping, illustrating heat transfer, saline cooling, and external forcing effects and aligning with physical expectations. The work lays groundwork for future developments in optimal control and inverse problems, with potential extensions to stochastic or enhanced bidomain formulations to capture additional physiological variability.

Abstract

This paper presents a nonlinear reaction-diffusion-fluid system that simulates radiofrequency ablation within cardiac tissue. The model conveys the dynamic evolution of temperature and electric potential in both the fluid and solid regions, along with the evolution of velocity within the solid region. By formulating the system that describes the phenomena across the entire domain, encompassing both solid and fluid phases, we proceed to an analysis of well-posedness, considering a broad class of right-hand side terms. The system involves parameters such as heat conductivity, kinematic viscosity, and electrical conductivity, all of which exhibit nonlinearity contingent upon the temperature variable. The mathematical analysis extends to establishing the existence of a global solution, employing the Faedo-Galerkin method in a three-dimensional space. To enhance the practical applicability of our theoretical results, we complement our study with a series of numerical experiments. We implement the discrete system using the finite element method for spatial discretization and an Euler scheme for temporal discretization. Nonlinear parameters are linearized through decoupling systems, as introduced in our continuous analysis. These experiments are conducted to demonstrate and validate the theoretical findings we have established.

Mathematical study of a new coupled electro-thermo radiofrequency model of cardiac tissue

TL;DR

The paper develops a coupled electro-thermo-fluid model for radiofrequency ablation in cardiac tissue, partitioning the domain into blood and tissue regions and enforcing interface conditions that couple Navier–Stokes with heat and potential equations. It proves the global existence of weak solutions in three dimensions using a Faedo–Galerkin approximation, uniform energy estimates, and compactness arguments, while addressing the nonlinear, temperature-dependent conductivities and electrode boundary conditions. Numerical validation is provided via two-dimensional finite element simulations with domain decomposition and decoupled time stepping, illustrating heat transfer, saline cooling, and external forcing effects and aligning with physical expectations. The work lays groundwork for future developments in optimal control and inverse problems, with potential extensions to stochastic or enhanced bidomain formulations to capture additional physiological variability.

Abstract

This paper presents a nonlinear reaction-diffusion-fluid system that simulates radiofrequency ablation within cardiac tissue. The model conveys the dynamic evolution of temperature and electric potential in both the fluid and solid regions, along with the evolution of velocity within the solid region. By formulating the system that describes the phenomena across the entire domain, encompassing both solid and fluid phases, we proceed to an analysis of well-posedness, considering a broad class of right-hand side terms. The system involves parameters such as heat conductivity, kinematic viscosity, and electrical conductivity, all of which exhibit nonlinearity contingent upon the temperature variable. The mathematical analysis extends to establishing the existence of a global solution, employing the Faedo-Galerkin method in a three-dimensional space. To enhance the practical applicability of our theoretical results, we complement our study with a series of numerical experiments. We implement the discrete system using the finite element method for spatial discretization and an Euler scheme for temporal discretization. Nonlinear parameters are linearized through decoupling systems, as introduced in our continuous analysis. These experiments are conducted to demonstrate and validate the theoretical findings we have established.
Paper Structure (20 sections, 3 theorems, 87 equations, 6 figures)

This paper contains 20 sections, 3 theorems, 87 equations, 6 figures.

Table of Contents

  1. Introduction and problem statement
  2. Background
  3. Electrothermal field in thermistor models
  4. Bio-heat distribution
  5. Blood flow model
  6. Electro-thermo-fluid model
  7. Contributions and related works
  8. Organization of the paper
  9. Notion of solution and main result
  10. Mathematical setting
  11. Variational formulation
  12. Proof of Theorem \ref{['theo1']}
  13. Faedo-Galerkin approximate solutions
  14. Basic a priori estimates
  15. Passage to the limit and concluding the proof of Theorem \ref{['theo1']}
  16. ...and 5 more sections

Key Result

Lemma 2.1

The trilinear form b is continuous on $\mathbf{H}_{0}^{1}(\Omega) \times\mathbf{H}_{0}^{1}(\Omega) \times\mathbf{H}_{0}^{1}(\Omega)$ and satisfies: Moreover, for all $\boldsymbol{w}, \boldsymbol{v}, \boldsymbol{\psi} \in \boldsymbol {\mathcal{H}}^{\boldsymbol{u}}_{\boldsymbol{0}},$ we have

Figures (6)

  • Figure 1: Illustration of radiofrequency ablation procedure within cardiac tissue, highlighting different regions in the domain. https://www.stopafib.org/procedures-for-afib/catheter-ablation/
  • Figure 2: Configuration geometry and boundaries conditions of the model.
  • Figure 3: Snapshot of evolution of the potentials at four time moments $t=\frac{T}{6}$, $\frac{T}{2}$, $\frac{3T}{4}$, $T$.
  • Figure 4: Test 1 : evolution of heats (column 1), velocity and pressure (column 2) at three time moments $t=\frac{T}{6}$ (line 1), $t= \frac{T}{2}$ (line 2) and $t=T$ (line 3).
  • Figure 5: Test 2 : evolution of heats (column 1), velocity and pressure (column 2) at three time moments $t=\frac{T}{6}$ (line 1), $t= \frac{T}{2}$ (line 2) and $t=T$ (line 3).
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 2.1: Properties of the trilinear form boyer2012mathematical
  • Definition 2.1
  • Theorem 2.1
  • Lemma 3.1