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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 (Honorary professor)

In this paper, we propose a monolithic algorithm for the numerical solution of theelectromechanics model of the left ventricle in the human heart. Our coupled modelintegrates the monodomain equation with the Bueno-Orovio minimal model for electrophysiologyand the Holzapfel-Ogden constitutive law for the passive mechanics ofthe myocardium; a distinguishing feature of our electromechanics model is the use ofthe active strain formulation for muscle contraction, which we exploit - for the firsttime in this context - by means of a transmurally variable active strain formulation. Weuse the Finite Element method for space discretization and Backward DifferentiationFormulas for time discretization, which we consider for both implicit and semi-implicitschemes. We compare and discuss the two schemes in terms of computational efficiencyas the semi-implicit scheme poses significant restrictions on the timestep size due tostability considerations, while the implicit scheme yields instead a nonlinear problem,which we solve by means of the Newton method. Emphasis is laid on preconditioningstrategy of the linear solver, which we perform by factorizing a block Gauss-Seidel preconditionerin combination with combination with parallel preconditioners for each ofthe single core models composing the integrated electromechanics model. We carry outseveral numerical simulations in the High Performance Computing framework for bothidealized and patient-specific left ventricle geometries and meshes, which we obtainby segmenting medical MRI images. We produce personalized pressure-volume loopsby means of the computational procedure, which we use to synthetically interpret andanalyze the outputs of the electromechanics model.
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Keywords heart modeling; coupled problem; electromechanics; monolithic algorithm; finite element method; preconditioner

Citation: A. Gerbi, L. Dedè, A. Quarteroni. A monolithic algorithm for the simulation of cardiac electromechanics in the human left ventricle. Mathematics in Engineering, 2018, 1(1): 1-46. doi: 10.3934/Mine.2018.1.1

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