A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle

Antonello Gerbi; Luca Dedè; Alfio Quarteroni; Antonello Gerbi; Luca Dedè; Alfio Quarteroni

doi:10.3934/Mine.2018.1.1

Mathematics in Engineering

2019, Volume 1, Issue 1: 1-37. doi: 10.3934/Mine.2018.1.1

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Research article

A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle

1.
Chair of Modelling and Scientific Computing (CMCS), Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Avenue Piccard, CH-1015, Lausanne, Switzerland
2.
MOX-Modeling and Scientific Computing, Mathematics Department, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy
3.
Institute of Mathematics,École Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland

Received: 25 April 2018 Accepted: 22 May 2018 Published: 28 August 2018

In this paper, we propose a monolithic algorithm for the numerical solution of the electromechanics model of the left ventricle in the human heart. Our coupled model integrates the monodomain equation with the Bueno-Orovio minimal model for electrophysiology and the Holzapfel-Ogden constitutive law for the passive mechanics of the myocardium; a distinguishing feature of our electromechanics model is the use of the active strain formulation for muscle contraction, which we exploit -- for the first time in this context -- by means of a transmurally variable active strain formulation. We use the Finite Element method for space discretization and Backward Differentiation Formulas for time discretization, which we consider for both implicit and semi-implicit schemes. We compare and discuss the two schemes in terms of computational efficiency as the semi-implicit scheme poses significant restrictions on the timestep size due to stability considerations, while the implicit scheme yields instead a nonlinear problem, which we solve by means of the Newton method. Emphasis is laid on preconditioning strategy of the linear solver, which we perform by factorizing a block Gauss-Seidel preconditioner in combination with combination with parallel preconditioners for each of the single core models composing the integrated electromechanics model. We carry out several numerical simulations in the High Performance Computing framework for both idealized and patient-specific left ventricle geometries and meshes, which we obtain by segmenting medical MRI images. We produce personalized pressure-volume loops by means of the computational procedure, which we use to synthetically interpret and analyze the outputs of the electromechanics model.
- heart modeling,
- coupled problem,
- electromechanics,
- monolithic algorithm,
- finite element method,
- preconditioner
Citation: Antonello Gerbi, Luca Dedè, Alfio Quarteroni. A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle[J]. Mathematics in Engineering, 2019, 1(1): 1-37. doi: 10.3934/Mine.2018.1.1

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Abstract

In this paper, we propose a monolithic algorithm for the numerical solution of the electromechanics model of the left ventricle in the human heart. Our coupled model integrates the monodomain equation with the Bueno-Orovio minimal model for electrophysiology and the Holzapfel-Ogden constitutive law for the passive mechanics of the myocardium; a distinguishing feature of our electromechanics model is the use of the active strain formulation for muscle contraction, which we exploit -- for the first time in this context -- by means of a transmurally variable active strain formulation. We use the Finite Element method for space discretization and Backward Differentiation Formulas for time discretization, which we consider for both implicit and semi-implicit schemes. We compare and discuss the two schemes in terms of computational efficiency as the semi-implicit scheme poses significant restrictions on the timestep size due to stability considerations, while the implicit scheme yields instead a nonlinear problem, which we solve by means of the Newton method. Emphasis is laid on preconditioning strategy of the linear solver, which we perform by factorizing a block Gauss-Seidel preconditioner in combination with combination with parallel preconditioners for each of the single core models composing the integrated electromechanics model. We carry out several numerical simulations in the High Performance Computing framework for both idealized and patient-specific left ventricle geometries and meshes, which we obtain by segmenting medical MRI images. We produce personalized pressure-volume loops by means of the computational procedure, which we use to synthetically interpret and analyze the outputs of the electromechanics model.

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