Youssef Belhamadia

Towards accurate numerical method for monodomain models using a realistic heart geometry

The simulation of cardiac electrophysiological waves are known to require extremely fine meshes, limiting the applicability of current numerical models to simplified geometries and ionic models. In this work, an accurate numerical method based on a time-dependent anisotropic remeshing strategy is presented for simulating three-dimensional cardiac electrophysiological waves. The proposed numerical method greatly reduces the number of elements and enhances the accuracy of the prediction of the electrical wave fronts. Illustrations of the performance and the accuracy of the proposed method are presented using a realistic heart geometry. Qualitative and quantitative results show that the proposed methodology is far superior to the uniform mesh methods commonly used in cardiac electrophysiology.

A Time-Dependent Adaptive Remeshing for Electrical Waves of the Heart

In this work, a time-dependent remeshing strategy and a numerical method are presented for the simulation of the action potential propagation of the human heart. The main purpose of these simulations is to accurately predict the depolarization-repolarization front position, which is essential to the understanding of the electrical activity in the myocardium. A bidomain model, which is commonly used for studying electrophysiological waves in the cardiac tissue, will be employed for the numerical simulations. Numerical results are enhanced by the introduction of an anisotropic remeshing strategy. The illustration of the performance and the accuracy of the proposed method are presented using a 2-D analytical solution and a test case with re-entrant waves.

Numerical prediction of freezing fronts in cryosurgery: Comparison with experimental results

Recent developments in scientific computing now allow to consider realistic applications of numerical modelling to medicine. In this work, a numerical method is presented for the simulation of phase change occurring in cryosurgery applications. The ultimate goal of these simulations is to accurately predict the freezing front position and the thermal history inside the ice ball which is essential to determine if cancerous cells have been completely destroyed. A semi-phase field formulation including blood flow considerations is employed for the simulations. Numerical results are enhanced by the introduction of an anisotropic remeshing strategy. The numerical procedure is validated by comparing the predictions of the model with experimental results.

Optimal control design for time-varying catalytic reactors: a Riccati equation-based approach

The linear quadratic (LQ) optimal control problem is studied for a partial differential equation model of a time-varying catalytic reactor. First, the dynamical properties of the linearised model are studied. Next, an LQ-control feedback is computed by using the corresponding operator Riccati differential equation, whose solution can be obtained via a related matrix Riccati partial differential equation. Finally, the designed controller is applied to the non-linear reactor system and tested numerically.

On the performance of anisotropic mesh adaptation for scroll wave turbulence dynamics in reaction–diffusion systems

Abstract Nonlinear reaction–diffusion systems are widely employed to study the spatio-temporal chaotic behavior that occurs in excitable media such as cardiac tissue where sufficiently strong perturbations can excite nonlinear propagating waves which can form spiral waves in two dimensions or scroll waves in three dimensions. The numerical simulation of these waves calls for grids that are extremely fine over the whole computational domain to accurately predict the trajectories and multiplication of wave fronts and therefore leads to huge computational challenges. Mesh adaptation methods can reduce the number of degrees of freedom required for a given accuracy but they also have a cost and it is not clear if they are competitive with respect to very fine uniform meshes. Previous mesh adaptation techniques applied to spatio-temporal chaotic behavior have been mostly limited to the two-dimensional case. The purpose of this paper is to explore the efficiency of a three-dimensional anisotropic finite element mesh adaptivity for simulating scroll wave turbulence. The computational efficiency of the proposed method is assessed using reference solutions obtained on a uniform refined mesh with more than 44 millions degrees of freedom. The proposed method reduces significantly the number of elements leading to huge saving in memory as well as in computational time. Examples of the dynamics of ventricular fibrillation in cardiac tissue will be presented illustrating the performance of the overall methodology.

Control of cardiac alternans in an electromechanical model of cardiac tissue

Electrical alternations in cardiac action potential duration have been shown to be a precursor to arrhythmias and sudden cardiac death. Through the mechanism of excitation-contraction coupling, the presence of electrical alternans induces alternations in the heart muscle contractile activity. Also, contraction of cardiac tissue affects the process of cardiac electric wave propagation through the mechanism of the so-called mechanoelectrical feedback. Electrical excitation and contraction of cardiac tissue can be linked by an electromechanical model such as the Nash-Panfilov model. In this work, we explore the feasibility of suppressing cardiac alternans in the Nash-Panfilov model which is employed for small and large deformations. Several electrical pacing and mechanical perturbation feedback strategies are considered to demonstrate successful suppression of alternans on a one-dimensional cable. This is the first attempt to combine electrophysiologically relevant cardiac models of electrical wave propagation and contractility of cardiac tissue in a synergistic effort to suppress cardiac alternans. Numerical examples are provided to illustrate the feasibility and the effects of the proposed algorithms to suppress cardiac alternans.

Recent Numerical Methods in Electrocardiology

Heart diseases are the leading cause of death in the world. Many questions have not yet been answered regarding the electrical waves propagation in cardiac tissue, and the mechanism of ventricular fibrillation that is produced by one or many spiral propagation waves of the excitation cardiac wall. Numerical modeling can play a crucial role and provides the necessary tools to answer some of these questions. However, the mathematical models, which give the best reflection of electrophysiological waves in cardiac tissue, are extremely complicated and present a significant computational challenges. The bidomain model is considered as the mathematical equations that have been used for simulating cardiac electrophysiological waves for many years (see Sundnes (2002), and Pierre (2006) and the reference therein). This model represents the cardiac tissue at a macroscopic scale by relating the transmembrane potential, the extracellular potential, and the ionic currents. The biodomain model consists of a system of two nonlinear partial differential equations coupled to a system of ordinary differential equations. From the numerical point of view, the model is computationally very expensive. The major difficulties are due to the computational grids size that must be very fine to get a realistic simulation of cardiac tissue. Indeed, the action potential is a wave with sharp depolarization and repolarization fronts and this wave travels across the whole computational domain calling for a very fine uniform mesh. One popular way of reducing the computational challenges of the bidomain model is the use of the monodomain model. This model considers a single nonlinear partial differential equation coupled with the same system of ordinary differential equations for the ionic currents. Although, it has been reported that the CPU requirements are reduced when simplifying the bidomain model to a monodomain model (see Sundnes et al. (2006)), both models still encounter computational difficulties because of the need for fine meshes and small time-steps. Many methods have been introduced in the literature to overcome these difficulties. The operator splitting is usually performed to separate the large non-linear system of ODEs and thus introduces subproblems easier to solve. A first-order (Godunov method) and a second-order (Strang method) accurate splitting technique can be employed. For more details the reader is referred to Sundnes et al. (2005), Lines, Buist, Grottum, Pullan, Sundnes & Tveito (2003); Lines, Grottum & Tveito (2003), and Weber Dos Santos et al. (2003)). To reduce the computational time at each time step, parallel computing techniques are used (see Colli Franzone & Pavarino (2004), Karpoukhin et al. (1995) and Weber dos Santos et al. (2004)). Several timestepping strategies have also been used, fully implicit ( Bourgault et al. (2003), and Murillo & Cai (2004)), and semi-implicit ( Franzone & Pavarino (2004), Ethier & Bourgault (2008))

Cardiac alternans annihilation by distributed mechano-electric feedback (MEF)

The presence of the electrical alternans induces, through the mechanism of the excitation-contraction coupling, an alternation in the heart muscle contractile activity. In this work, we demonstrate the cardiac alternans annihilation by applied mechanical perturbation. In particular, we address annihilation of alternans in realistic heart size tissue by considering ionic currents suggested by Luo-Rudy-1 (LR1) model, in which the control algorithm involves a combined electrical boundary pacing control and a spatially distributed calcium based control which perturbs the calcium in the cells. Complimentary to this, we also address a novel mechanism of alternans annihilation which uses a Nash Panfilov model coupled with the stress equilibrium equations. The coupled model includes an additional variable to represent the active stress which defines the mechanical properties of the tissue.

Numerical computations of cardiac AP using level set based geometries

This article proposes two avenues to help improve the realism of numerical computations for cardiac electrophysiology while maintaining manageable computational resources. We first propose an asymptotic analysis to adjust the parameters and use a simple two-variable ionic model to reproduce the main characteristics of the cardiac action potential (AP) in various myocardial regions. This ionic model is embedded in the bidomain model that is used to propagate the AP in the heart. Our second contribution is a finite element method that couples the heart with the torso in a single variational formulation and allows non body fitted meshes at the interface between the myocardium and the torso/ventricle cavities. This interface is described through a level-set function obtained from the segmentation of patient medical images. Using a 2D test case, we compare the use of body-fitted and non body-fitted meshes and analyze the impact of both approaches on the accuracy of the solutions, including an anisotropic mesh adaptation strategy.