Fehrenbach, Jérôme

Weynans, Lisl

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HAL Id: hal-03913733https://hal.inria.fr/hal-03913733Submitted on 27 Dec 2022HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Activation Sites Estimation using a Fast AlgorithmBased on an Eikonal EquationJérôme Fehrenbach, Lisl WeynansTo cite this version:Jérôme Fehrenbach, Lisl Weynans. Activation Sites Estimation using a Fast Algorithm Based on anEikonal Equation. Computing in Cardiology: CINC 2022, Sep 2022, Tampere, Finland. ￿hal-03913733￿Activation Sites Estimation using a Fast AlgorithmBased on an Eikonal EquationJérôme Fehrenbach, Lisl Weynans -- Univ. Toulouse, IMT and Univ. Bordeaux, Inria, IHU Liryc, FranceContext and objectivesI Classical algorithms for ECGI solve at each time instant an staticand extremely ill-posed inverse problem, requiring an adequateregularization strategy (or instance Tikhonov)I An alternative way of regularizing the inverse problem is toconsider information from other time steps, taking into accountthat electric potentials in the heart are obtained by thepropagation of an electric wave.I Our objectives: design a fast algorithm based on a simpleactivation front model to retrieve significant features of the ECGIinverse problemPropagation modelI We consider in the present work only the surface of the heartI Anisotropic eikonal equation: simplified model of electricalactivation front in asymptotic regime:{‖∇T‖2D(x) = 1,T(x0i ) = τi, i = 1 . . . s,with. T depolarization time of heart surface,. D(x) tensor quantity describing the anisotropic conduction. τi activation times at sources x0i (earliest activation sites)⇒ parameters elevant in the context of EGCII T(x) is also the length of the shortest path joining the sources to xFigure: Example of solution of eikonal equation with anisotropic conductionMethodsI Iterative algorithm to minimize misfit between observations andpredictions of the modelI Cost functionJ(x, τ ) = 12‖G(τ + φx)− gOBS‖2Y.with G observation operator (on heart or torso surface), x sourceslocation, gOBS observations with noiseI Least square minimization with Gauss Newton algorithmI Sensitivity of cost function with respect to source locations andactivation times depends on Logarithmic map:if γ is a unique shortest geodesic path from point x to point y thenLogx(y) = γ′(0) = initial direction of γI Vector Heat Method: efficient computational method to computeLogarithmic map, avoiding to compute all geodesicsFigure: Illustration of Log map from et alTest case: Localization of 2 sources and metric estimationFigure: Left: evolution of the cost function J. Right: evolution of the distance to thesolution ‖xk − x∗‖.Figure: Left: location of the successive iterates xk (red) and the true source point x∗(blue). Right: evolution of the parameters of the metric.Test case: mimicking an ECG inversionFigure: Left: configuration of the torso (electrodes in red) and the heart surface(visible by transparency). Right: location of the successive iterates xk (red) and thetrue source point x∗ (blue)Figure: Left: evolution of the cost function J. Right: evolution of the distance to thesolution ‖xk − x∗‖.ConclusionI Efficient new method to estimate sources and activation times inthe eikonal equation on a surface⇒ applicable to ECGI.I In case of complex or pathological propagation patterns, need toextend the method to a cardiac volume.References[1] J. Fehrenbach, L. Weynans, Source and metric estimation in theeikonal equation using optimization on a manifold, under revision[2] Grandits et al, GEASI: Geodesic-based Earliest Activation SitesIdentification in cardiac models, Int. J. Numer. Meth. BiomedEngng, 2021[3] N. Sharp et al, The vector heat method. ACM Transactions onGraphics, 2019.

Activation Sites Estimation using a Fast Algorithm Based on an Eikonal Equation

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Activation Sites Estimation using a Fast Algorithm Based on an Eikonal Equation

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