Abstract Electrocardiographic imaging (ECGi) aims to reconstruct cardiac electrical activity from noninvasive measurements of torso surface potentials. This inverse problem is commonly formulated as a PDE-constrained optimization problem relying on a source model. One common way to overcome its severe ill-posedness is to add a regularization term to the cost function to be minimized in the optimization problem. Another way consists in adding constraints in the optimization problem, for instance based on physical informations or modeling. Following this idea, in this work, we introduce a novel source model, named the epicardial model (EPI), which provides a heuristic three-dimensional extension of a previously proposed depth-averaged two-dimensional model. This new model is formulated in the torso domain only and couples a bidomain-like surface equation on the heart with the standard Laplace equation in the torso. Thus, it can naturally be substituted into standard torso-based clinical workflows while offering more possibilities for incorporating physiologically relevant prior information on the modeled EPI surface. The EPI is first validated numerically against reference bidomain simulations. While amplitude discrepancies are observed, the model is able to reproduce the spatial distribution of the electrical potentials. The EPI is then evaluated within an ECGi reconstruction framework. Two EPI-constrained optimization formulations are considered: one using classical Tikhonov regularization and another using a physiological total variation (TV) regularization applied to the transmembrane voltage. Reconstructions are compared with those obtained from the classical Laplace-based Cauchy formulation. Without addition of information, the proposed EPI based framework yields reconstruction of comparable accuracy to standard approaches despite the presence of modeling approximations. The incorporation of a prior through the TV regularization, allowed by the model, improves signals reconstruction and reduces artificial lines of block in activation maps. These results highlight the potential of the EPI as a physiologically enriched alternative to classical ECGi formulations.

We study the convergence of a Cartesian method for elliptic problems with immersed interfaces. This method is based on additional unknowns located on the interface, used to express the jump conditions across the interface and discretize the elliptic operator in each subdomain separately. It is numerically second-order accurate in L^∞ -norm. We prove the convergence of the method in two cases: the original second-order method in one dimension, and a first-order version in two dimensions. The proof of convergence takes advantage of a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix. More precisely, we obtain estimates for the sums of the coefficients of several blocks of the inverse matrix. Associated to the consistency error, which has different leading orders throughout the domain, these estimates lead to the convergence results. The methodology exposed in the article allows to take into account the effects of different orders of approximation errors across the domain and their effective influence on the total convergence order.

The inverse problem of cardiac electrophysiology is notoriously ill-posed, but is nonetheless extremely useful. In particular, it is difficult to reconstruct the transmembrane voltage in the volume of the heart, since an infinite dimensional space of transmembrane voltages can produce the same observation on the body surface. A widely used alternative is to consider only the outer surface of the heart, the epicardium, and solve a Cauchy problem for the Laplace equation in the torso domain. However, this approach only allows reconstruction of the extracellular potential on the epicardium, thus missing the information contained in the volume of the heart and in the transmembrane voltage. We propose a new methodology for reconstructing activation maps from torso surface data, which incorporates information from the myocardial volume, while solving a surface problem on the heart. We formulate a static forward model, derived from the bidomain model, by averaging equations in the heart. The averaged equations are coupled with the usual Laplace equations in the surrounding domains. For solving the inverse problem, this << depth-averaged >> forward model is used as a constraint in an optimal control problem that allows to recover depth-averaged transmembrane voltage and extracellular potential in the heart, corresponding to observations on the body surface. Activation maps are computed by post-processing the recovered signals in the heart. This method retains the ability to include the interactions between the heart, torso and blood cavities, while maintaining the simplicity of the usual inverse procedure. The inverse problem is solved with a simple linear system, using the Lagrangian formalism, and thus in a single iteration. We emphasize on the post-processing techniques for recovering activation maps. We observe that using a threshold on the transmembrane voltage allows to recover smoother and more accurate activation maps than with a maximal deflection method on the extracellular potential.

We briefly present a new epicardial source model for the inverse problem of cardiac electrocardiography. It couples a surface equation on the epicardium with the three-dimensional torso, while incorporating heart-torso fluxes and possible anisotropic conductivities in the intracellular and extracellular domains. As this capability is new for a surface model, we investigate the impact of inserting epicardial fiber directions into the surface source model on inverse reconstructions, compared with isotropy-based reconstructions. Our results show that the epicardial fibers degrade the epicardial model based ECGi reconstructions, in particular the transmembrane voltage, whereas isotropic reconstructions demonstrate significantly higher accuracy for the extracellular potential and the transmembrane voltage. We believe that, due to interactions and compensations within the myocardial volume, the solutions of the bidomain model taken on a surface are more similar to isotropic solutions than strongly anisotropic ones. In this case, taking an isotropic assumption in a surface model based ECGi is a relevant choice.

This work studies waves interacting with partially immersed objects moving freely in the vertical direction, and waves described by the one-dimensional Boussinesq-Abbott system. This problem can be reduced to a transmission problem, with transmission conditions involving the resolution of coupled forced ODEs satisfied by the vertical displacement of the object and the average discharge of the fluid under the object. We propose a new extended formulation of the problem that provides the value of the surface elevation at the two contact points. This formulation simplifies the numerical computation of transmission conditions and the discretization of the fluid equations in cells near the object. Based on this formulation, we propose a second order-scheme involving a generalization of the Mac-Cormack scheme with nonlocal flux and a source term. Several explicit solutions are derived to validate this scheme. They can serve as benchmark for future codes.

Electrocardiographic imaging non-invasively reconstructs activation maps of the heart from temporal body surface potential maps by post-processing solutions of an inverse problem. Typically, activation times are detected through the maximal deflection of the temporal or spatial derivative of recovered extracellular or transmembrane potentials. However, this method can introduce artificial lines of block in the map, falsely indicating a pathology. Consequently, several complex algorithms have been developed in an attempt to smooth activation maps while preserving true discontinuities. We propose a straightforward method for computing activation maps from recovered transmembrane voltages, wherein activation time is defined as the first time a predefined threshold is crossed. We evaluate this thresholdbased method against traditional deflection-based methods using simulated data, following the approach of Shuler et al. [1]. Our findings indicate that the threshold method, when combined with classical Tikhonov regularization, produces smooth activation maps even in the presence of true lines of block. Given the variability in performance of deflectionbased methods compared to those reported in [1], we emphasize the difficulty of establishing a universally effective post-processing method for computing activation maps.

Objective: Noninvasive electrocardiographic imaging (ECGI) reconstructs cardiac electrical activity from body surface potential measurements. However, current methods have demonstrated inaccuracies in reconstructing sinus rhythm, and in particular breakthrough sites. This study aims to combine existing inverse algorithms, making the most of their advantages while minimizing their limitations. Method: The “patchwork method” (PM) combines two classical numerical methods for ECGI: the method of fundamental solutions (MFS) and the finite-element method (FEM). We assume that the method with the smallest residual in the predicted torso potentials, computed using the boundary element method (BEM), provides the most accurate solution. The PM selects for each heart node and time step the method whose estimated reconstruction error is smallest. The performance of the PM was evaluated using simulated ectopic and normal ventricular beats. Results: Cardiac potentials and activation maps obtained with the PM (CC = 0.63 $\pm$ 0.01 and 0.61 $\pm$ 0.05 respectively) were more accurate than MFS (CC = 0.61 $\pm$ 0.01 and 0.48 $\pm$ 0.05 respectively), FEM (CC = 0.58 $\pm$ 0.01 and 0.51 $\pm$ 0.02 respectively) or BEM (CC = 0.57 $\pm$ 0.02 and 0.49 $\pm$ 0.02 respectively). The PM also located all epicardial breakthrough sites, whereas the traditional numerical methods usually missed one. Furthermore, the PM showed its robustness and stability in the presence of Gaussian noise added to the torso potentials. Conclusion: The PM overcomes some of the limitations of classical numerical methods, improving the accuracy of mapping important features of activation during sinus rhythm and paced beats. Significance: This novel method for optimizing ECGI solutions opens a new avenue for improving not only ECGI but also other inverse problems.

The goal of this work is to study waves interacting with partially immersed objects allowed to move freely in the vertical direction, and in a regime in which the propagation of the waves is described by the one dimensional Boussinesq-Abbott system. The problem can be reduced to a transmission problem for this Boussinesq system, in which the transmission conditions between the components of the domain at the left and at the right of the object are determined through the resolution of coupled forced ODEs in time satisfied by the vertical displacement of the object and the average discharge in the portion of the fluid located under the object. We propose a new extended formulation in which these ODEs are complemented by two other forced ODEs satisfied by the trace of the surface elevation at the contact points. The interest of this new extended formulation is that the forcing terms are easy to compute numerically and that the surface elevation at the contact points is furnished for free. Based on this formulation, we propose a second order scheme that involves a generalization of the MacCormack scheme with nonlocal flux and a source term, which is coupled to a second order Heun scheme for the ODEs. In order to validate this scheme, several explicit solutions for this wave-structure interaction problem are derived and can serve as benchmark for future codes. As a byproduct, our method provides a second order scheme for the generation of waves at the entrance of the numerical domain for the Boussinesq-Abbott system.

In order to establish a link between the activation map and the body surface signals, we want to compute body potentials from activation maps and a frontlike approximation of the activation of the heart.To do so, two formulations that map the extracardiac potential from the transmembrane voltage naturally emerge from the bidomain model.Either the extracellular/extracardiac potential solves a Laplace equation with discontinuous conductivity coefficient and ionic current as a source (Source Formulation F1); or the quasi-stationary electrical balance between the intra-and extracellular fields (Balance Formulation F2).In this work, we compare F1 and F2, to determine which formulation is the most relevant to use.We compute reference activation map ψ, transmembrane voltage v and body surface map u with a bidomain 2D code.We design two alternative shapes ṽ for a frontlike approximation of v. Afterwards, two extracellular / extracardiac potentials are computed from the activation time ψ and ṽ, using the two different formulations.Then the extracardiac potentials solutions of F1 and F2 are compared respectively to the solution of the bidomain u.Results show that the Balance Formulation F2 is robust to the input data (ṽ and ψ).On the contrary, the Source Formulation F1 is very unstable and generates very large errors on the body surface map.

We propose an immersed boundary scheme for the numerical resolution of the Complete Electrode Model in Electrical Impedance Tomography, that we use as a main ingredient in the resolution of inverse problems in medical imaging. Such method allows to use a Cartesian mesh without accurate discretization of the boundary, which is useful in situations where the boundary is complicated and/or changing. We prove the convergence of our method, and illustrate its efficiency with two dimensional direct and inverse problems.

We address the estimation of the source(s) location in the eikonal equation on a Riemann surface, as well as the determination of the metric when it depends on a few parameters. The available observations are the arrival times or are obtained indirectly from the arrival times by an observation operator, this frame is intended to describe electro-cardiographic imaging. The sensitivity of the arrival times is computed from \begin{document}$ {{{\rm{Log}}}}_x $\end{document} the log map wrt to the source \begin{document}$ x $\end{document} on the surface. The \begin{document}$ {{{\rm{Log}}}}_x $\end{document} map is approximated by solving an elliptic vectorial equation, using the Vector Heat Method. The \begin{document}$ L^2 $\end{document}-error function between the model predictions and the observations is minimized using Gauss-Newton optimization on the Riemann surface. This allows to obtain fast convergence. We present numerical results, where coefficients describing the metric are also recovered like anisotropy and global orientation.

Torso surface and ventricular epicardial potentials were recorded simultaneously in anesthetized, closed-chest pigs $(n=5)$ during sinus rhythm. Activation times were estimated from recorded torso potentials using three classical ECGI methods and a new method called the Patchwork Method (PM), which locally selects the optimal ECGI method and has demonstrated its efficiency with simulated data. The aim of this study was to evaluate the Patchwork method using experimental data in sinus rhythm. By comparing the classic ECGI reconstructions to recorded epicardial activation mapping, several inaccuracies in the ECGI maps are highlighted in this study. This involved inaccuracies in reconstructing activating maps, locating breakthrough sites and the production of artificial lines of block. However, the PM overcomes these restrictions, demonstrating its abilities to accurately reconstruct activation maps $(CC=0.90\ \ [0.86;0.92]$ and $RE=0.20$ [0.19; 0.24]) and localize epicardial breakthrough sites $(LE=17.16\ \ [8.87;22.14])$. Furthermore, it reduced the frequency of artificial lines of block (2 of 5 pig hearts).

An Eulerian method to numerically solve incompressible bifluid problems with high density ratio is presented. This method can be considered as an improvement of the Ghost Fluid method, with the specificity of a sharp second-order numerical scheme for the spatial resolution of the discontinuous elliptic problem for the pressure. The Navier–Stokes equations are integrated in time with a fractional step method based on the Chorin scheme and discretized in space on a Cartesian mesh. The bifluid interface is implicitly represented using a level-set function. The advantage of this method is its simplicity to implement in a standard monofluid Navier–Stokes solver while being more accurate and conservative than other simple classical bifluid methods. The numerical tests highlight the improvements obtained with this sharp method compared to the reference standard first-order methods.

Noninvasive electrocardiographic imaging (ECGI) pro-vides real-time panoramic images of epicardial electri-cal activity from potential measurements on the torso surface. This non-invasive imaging modality can be-come a powerful clinical tool to help understand themechanisms underlying many cardiac diseases, and to define the appropriate treatment. ECGI is mathematically represented by a Cauchy problem for the Laplace equation

We present a new method for the numerical implementation of generating boundary conditions for a one dimensional Boussinesq system. This method is based on a reformulation of the equations and a resolution of the dispersive boundary layer that is created at the boundary when the boundary conditions are non homogeneous. This method is implemented for a simple first order finite volume scheme and validated by several numerical simulations. Contrary to the other techniques that can be found in the literature, our approach does not cause any increase in computational time with respect to periodic boundary conditions.

Noninvasive electrocardiographic imaging (ECGI) methods are known to produce artificial lines of block in healthy tissue. However, it remains unclear if ECGI can detect regions of slow conduction in damaged hearts. The aim of this study was to develop and test a method to detect the presence of slow conduction zones using ECGI. Activation times were estimated from simulated electrocardiograms using two classical ECGI methods and a new method called the Patchwork Method (PM), which locally selects the optimal ECGI method. Five different methods (maximum, minimum, mean, nearest neighbor and random) to compute activation time gradients were then tested to pinpoint the slow conduction zones. Overall, the local maximum and mean gradients both accurately located the zones of tissue damage. While the classical ECGI methods did not identify any of these regions, the Patchwork Method succeeded in identifying 5 of the 6 slow conduction zones. This novel approach thus overcomes some of the limitations of classic ECGI methods to detect regions of slow conduction.

Despite being relevant in many natural and industrial processes, suspensions of nonspherical particles have been largely underinvestigated compared with the extensive analyses made on the gravity-driven motions of spherical particles. One of the main reasons for this disparity is the difficulty of accurately correcting the short-range hydrodynamic forces and torques acting on complex particles. These effects, also known as lubrication, are essential to the suspension of the particles and are usually poorly captured by direct numerical simulation of particle-laden flows. In this article, we propose a partitioned volume penalization-discrete element method solver, which estimates the unresolved hydrodynamic forces and torques. Corrections are made locally on the surface of the interacting particles without any assumption on the particle global geometry. Numerical validations have been made using ellipsoidal particles immersed in an incompressible Navier-Stokes flow.

The lubrication forces are short-range hydrodynamic interactions essential to describe suspension of the particles. Usually, they are underestimated in direct numerical simulations of particle-laden flows. In this paper, we propose a lubrication model for a coupled volume penalization method and discrete element method solver that estimates the unresolved hydrodynamic forces and torques in an incompressible Navier-Stokes flow. Corrections are made locally on the surface of the interacting particles without any assumption on the global particle shape. The numerical model has been validated against experimental data and performs as well as existing numerical models that are limited to spherical particles.

We prove in this paper the second-order super-convergence in \(L^{\infty }\)-norm of the gradient for the Shortley–Weller method. Indeed, this method is known to be second-order accurate for the solution itself and for the discrete gradient, although its consistency error near the boundary is only first-order. We present a proof in the finite-difference spirit, using a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix. The proof is based on a discrete Poisson equation for the discrete gradient, with second-order accurate Dirichlet boundary conditions. The advantage of this finite-difference approach is that it can provide pointwise convergence results depending on the local consistency error and the location on the computational domain.