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Propp A, Gizzi A, Levrero-Florencio F, Ruiz-Baier R. An orthotropic electro-viscoelastic model for the heart with stress-assisted diffusion. Biomech Model Mechanobiol 2020; 19:633-659. [PMID: 31630280 PMCID: PMC7105452 DOI: 10.1007/s10237-019-01237-y] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2019] [Accepted: 10/09/2019] [Indexed: 12/21/2022]
Abstract
We propose and analyse the properties of a new class of models for the electromechanics of cardiac tissue. The set of governing equations consists of nonlinear elasticity using a viscoelastic and orthotropic exponential constitutive law, for both active stress and active strain formulations of active mechanics, coupled with a four-variable phenomenological model for human cardiac cell electrophysiology, which produces an accurate description of the action potential. The conductivities in the model of electric propagation are modified according to stress, inducing an additional degree of nonlinearity and anisotropy in the coupling mechanisms, and the activation model assumes a simplified stretch-calcium interaction generating active tension or active strain. The influence of the new terms in the electromechanical model is evaluated through a sensitivity analysis, and we provide numerical validation through a set of computational tests using a novel mixed-primal finite element scheme.
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Affiliation(s)
- Adrienne Propp
- Mathematical Institute, University of Oxford, A. Wiles Building, Woodstock Road, Oxford, OX2 6GG United Kingdom
| | - Alessio Gizzi
- Nonlinear Physics and Mathematical Modeling Laboratory, Department of Engineering, University Campus Bio-Medico, Rome, Italy
| | | | - Ricardo Ruiz-Baier
- Mathematical Institute, University of Oxford, A. Wiles Building, Woodstock Road, Oxford, OX2 6GG United Kingdom
- Laboratory of Mathematical Modelling, Institute of Personalised Medicine, Sechenov University, Moscow, Russian Federation
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Levrero-Florencio F, Margara F, Zacur E, Bueno-Orovio A, Wang Z, Santiago A, Aguado-Sierra J, Houzeaux G, Grau V, Kay D, Vázquez M, Ruiz-Baier R, Rodriguez B. Sensitivity analysis of a strongly-coupled human-based electromechanical cardiac model: Effect of mechanical parameters on physiologically relevant biomarkers. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2020; 361:112762. [PMID: 32565583 PMCID: PMC7299076 DOI: 10.1016/j.cma.2019.112762] [Citation(s) in RCA: 26] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/04/2023]
Abstract
The human heart beats as a result of multiscale nonlinear dynamics coupling subcellular to whole organ processes, achieving electrophysiologically-driven mechanical contraction. Computational cardiac modelling and simulation have achieved a great degree of maturity, both in terms of mathematical models of underlying biophysical processes and the development of simulation software. In this study, we present the detailed description of a human-based physiologically-based, and fully-coupled ventricular electromechanical modelling and simulation framework, and a sensitivity analysis focused on its mechanical properties. The biophysical detail of the model, from ionic to whole-organ, is crucial to enable future simulations of disease and drug action. Key novelties include the coupling of state-of-the-art human-based electrophysiology membrane kinetics, excitation-contraction and active contraction models, and the incorporation of a pre-stress model to allow for pre-stressing and pre-loading the ventricles in a dynamical regime. Through high performance computing simulations, we demonstrate that 50% to 200% - 1000% variations in key parameters result in changes in clinically-relevant mechanical biomarkers ranging from diseased to healthy values in clinical studies. Furthermore mechanical biomarkers are primarily affected by only one or two parameters. Specifically, ejection fraction is dominated by the scaling parameter of the active tension model and its scaling parameter in the normal direction ( k ort 2 ); the end systolic pressure is dominated by the pressure at which the ejection phase is triggered ( P ej ) and the compliance of the Windkessel fluid model ( C ); and the longitudinal fractional shortening is dominated by the fibre angle ( ϕ ) and k ort 2 . The wall thickening does not seem to be clearly dominated by any of the considered input parameters. In summary, this study presents in detail the description and implementation of a human-based coupled electromechanical modelling and simulation framework, and a high performance computing study on the sensitivity of mechanical biomarkers to key model parameters. The tools and knowledge generated enable future investigations into disease and drug action on human ventricles.
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Affiliation(s)
- F. Levrero-Florencio
- Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
- Corresponding authors.
| | - F. Margara
- Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
| | - E. Zacur
- Department of Engineering Science, University of Oxford, Oxford OX3 7DQ, United Kingdom
| | - A. Bueno-Orovio
- Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
| | - Z.J. Wang
- Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
| | - A. Santiago
- Barcelona Supercomputing Center – Centro Nacional de Supercomputación, Barcelona 08034, Spain
| | - J. Aguado-Sierra
- Barcelona Supercomputing Center – Centro Nacional de Supercomputación, Barcelona 08034, Spain
| | - G. Houzeaux
- Barcelona Supercomputing Center – Centro Nacional de Supercomputación, Barcelona 08034, Spain
| | - V. Grau
- Department of Engineering Science, University of Oxford, Oxford OX3 7DQ, United Kingdom
| | - D. Kay
- Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
| | - M. Vázquez
- Barcelona Supercomputing Center – Centro Nacional de Supercomputación, Barcelona 08034, Spain
- ELEM Biotech, Spain
| | - R. Ruiz-Baier
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
- Universidad Adventista de Chile, Casilla 7-D, Chillan, Chile
| | - B. Rodriguez
- Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
- Corresponding authors.
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Landajuela M, Vergara C, Gerbi A, Dedè L, Formaggia L, Quarteroni A. Numerical approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2018; 34:e2984. [PMID: 29575751 DOI: 10.1002/cnm.2984] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2017] [Revised: 02/02/2018] [Accepted: 03/11/2018] [Indexed: 06/08/2023]
Abstract
In this work, we consider the numerical approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network. The mathematical model couples the 3D elastodynamics and bidomain equations for the electrophysiology in the myocardium with the 1D monodomain equation in the Purkinje network. For the numerical solution of the coupled problem, we consider a fixed-point iterative algorithm that enables a partitioned solution of the myocardium and Purkinje network problems. Different levels of myocardium-Purkinje network splitting are considered and analyzed. The results are compared with those obtained using standard strategies proposed in the literature to trigger the electrical activation. Finally, we present a numerical study that, although performed in an idealized computational domain, features all the physiological issues that characterize a heartbeat simulation, including the initiation of the signal in the Purkinje network and the systolic and diastolic phases.
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Affiliation(s)
- Mikel Landajuela
- MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy
| | - Christian Vergara
- MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy
| | - Antonello Gerbi
- Chair of Modelling and Scientific Computing, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Route Cantonale, Lausanne, CH-1015, Switzerland
| | - Luca Dedè
- MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy
| | - Luca Formaggia
- MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy
| | - Alfio Quarteroni
- MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy
- Chair of Modelling and Scientific Computing, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Route Cantonale, Lausanne, CH-1015, Switzerland
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