Reduced order model in cardiac electrophysiology with approximated Lax pairs

Gerbeau, Jean-Frédéric; Lombardi, Damiano; Schenone, Elisa

doi:10.1007/s10444-014-9393-9

Cited by 19 publications

(25 citation statements)

References 25 publications

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“…We consider the initial datum u 0 to be positive on a band of the endocardium, representing the initial stimulus provided by the heart conducting system. The most important parameters, considered in accordance with [25], [41], are reported in Table 1. Table 1: Physical coefficients…”

Section: Finite Element Approximation Of Initial and Boundary Value Pmentioning

confidence: 99%

An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

Beretta¹,

Cavaterra²,

Cerutti³

et al. 2017

Inverse Problems

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In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, timedependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of an inhomogeneity ω (where the coefficients of the equation are altered) located inside a domain Ω starting from observations of the potential on the boundary ∂Ω. Such a problem is related to the detection of myocardial ischemic regions, characterized by severely reduced blood perfusion and consequent lack of electric conductivity. In the first part of the paper we provide an asymptotic formula for electric potential perturbations caused by internal conductivity inhomogeneities of low volume fraction, extending the results published in [7] to the case of three-dimensional, parabolic problems. In the second part we implement a reconstruction procedure based on the topological gradient of a suitable cost functional. Numerical results obtained on an idealized three-dimensional left ventricle geometry for different measurement settings assess the feasibility and robustness of the algorithm.

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Section: Finite Element Approximation Of Initial and Boundary Value Pmentioning

confidence: 99%

An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

Beretta¹,

Cavaterra²,

Cerutti³

et al. 2017

Inverse Problems

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show abstract

“…As a matter of fact, when following an optimize-then-discretize approach, we need to evaluate the Monodomain system and its adjoint, forward and backward in time, at each minimization iteration. This led to the introduction of model reduction techniques, based either on a Proper Orthogonal Decomposition (POD) paradigm [26,28,29] or the Lax-pairs [27]. The POD paradigm requires the offline computation of snapshots for different values of the parameters.…”

Section: The Monodomain Inverse Conductivity Problemmentioning

confidence: 99%

“…However, the efficiency of such procedures needs to be properly addressed, as the computational cost of the iterative mismatch minimization is generally high, especially when dealing with real geometries. This problem has been promptly recognized as a bottleneck, and several Reduced-Order Models (ROMs) have been investigated [26,27,28,29,13]. The proposed approaches rely on the construction of a surrogate, as a combination of basis functions generally built moving from previous solutions (called "snapshots") for a predetermined set of values for the parameters.…”

Section: Introductionmentioning

confidence: 99%

Efficient estimation of cardiac conductivities: A proper generalized decomposition approach

Barone

Carlino

Gizzi

et al. 2020

Journal of Computational Physics

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While the potential groundbreaking role of mathematical modeling in electrophysiology has been demonstrated for therapies like cardiac resynchronization or catheter ablation, its extensive use in clinics is prevented by the need of an accurate customized conductivity identification. Data assimilation techniques are, in general, used to identify parameters that cannot be measured directly, especially in patient-specific settings. Yet, they may be computationally demanding. This conflicts with the clinical timelines and volumes of patients to analyze. In this paper, we adopt a model reduction technique, developed by F. Chinesta and his collaborators in the last 15 years, called Proper Generalized Decomposition (PGD), to accelerate the estimation of the cardiac conductivities required in the modeling of the cardiac electrical dynamics. Specifically, we resort to the Monodomain Inverse Conductivity Problem (MICP) deeply investigated in the literature in the last five years. We provide a significant proof of concept that PGD is a breakthrough in solving the MICP within reasonable timelines. As PGD relies on the offline/online paradigm and does not need any preliminary knowledge of the high-fidelity solution, we show that the PGD online phase estimates the conductivities in real-time for both two-dimensional and three-dimensional cases, including a patient-specific ventricle.

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“…A preliminary version of local ROM has been introduced in [31], however performing an ad-hoc partition of snapshots in the parameter space, without relying on a general clustering procedure. An alternative strategy, introduced in [32,33], is based on the Lax-Pairs approach: here the basis functions are moved in time according to the traveling front. This approach entails additional online costs and, so far, is limited to two-dimensional problems.…”

Section: The Content Of This Paper and Comparison With Other Existingmentioning

confidence: 99%

Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method

Pagani

Manzoni

Quarteroni

2018

Computer Methods in Applied Mechanics and Engineering

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The efficient solution of coupled PDEs/ODEs problems arising in cardiac electrophysiology is of key importance whenever interested to study the electrical behavior of the tissue for several instances of relevant physical and/or geometrical parameters. This poses significant challenges to reduced order modeling (ROM) techniques-such as the reduced basis method-traditionally employed when dealing with the repeated solution of parameter dependent differential equations. Indeed, the nonlinear nature of the problem, the presence of moving fronts in the solution, and the high sensitivity of this latter to parameter variations, make the application of standard ROM techniques very problematic. In this paper we propose a local ROM built through a k-means clustering in the state space of the snapshots for both the solution and the nonlinear term. Several comparisons among alternative local ROMs on a benchmark test case show the effectivity of the proposed approach. Finally, the application to a parametrized problem set on an idealized leftventricle geometry shows the capability of the proposed ROM to face complex problems.

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Reduced order model in cardiac electrophysiology with approximated Lax pairs

Cited by 19 publications

References 25 publications

An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

Efficient estimation of cardiac conductivities: A proper generalized decomposition approach

Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method

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