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Tikenoğullar i OZ, Peirlinck M, Chubb H, Dubin AM, Kuhl E, Marsden AL. Effects of cardiac growth on electrical dyssynchrony in the single ventricle patient. Comput Methods Biomech Biomed Engin 2024; 27:1011-1027. [PMID: 37314141 PMCID: PMC10719423 DOI: 10.1080/10255842.2023.2222203] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2022] [Revised: 04/27/2023] [Accepted: 05/04/2023] [Indexed: 06/15/2023]
Abstract
Single ventricle patients, including those with hypoplastic left heart syndrome (HLHS), typically undergo three palliative heart surgeries culminating in the Fontan procedure. HLHS is associated with high rates of morbidity and mortality, and many patients develop arrhythmias, electrical dyssynchrony, and eventually ventricular failure. However, the correlation between ventricular enlargement and electrical dysfunction in HLHS physiology remains poorly understood. Here we characterize the relationship between growth and electrophysiology in HLHS using computational modeling. We integrate a personalized finite element model, a volumetric growth model, and a personalized electrophysiology model to perform controlled in silico experiments. We show that right ventricle enlargement negatively affects QRS duration and interventricular dyssynchrony. Conversely, left ventricle enlargement can partially compensate for this dyssynchrony. These findings have potential implications on our understanding of the origins of electrical dyssynchrony and, ultimately, the treatment of HLHS patients.
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Affiliation(s)
- O. Z. Tikenoğullar i
- Department of Mechanical Engineering, Stanford University, Stanford, California, USA
| | - M. Peirlinck
- Department of Biomechanical Engineering, Delft University of Technology, Delft, Netherlands
| | - H. Chubb
- Department of Pediatrics (Cardiology), Stanford University, Stanford, California, USA
| | - A. M. Dubin
- Department of Pediatrics (Cardiology), Stanford University, Stanford, California, USA
| | - E. Kuhl
- Department of Mechanical Engineering, Stanford University, Stanford, California, USA
| | - A. L. Marsden
- Department of Mechanical Engineering, Stanford University, Stanford, California, USA
- Department of Pediatrics (Cardiology), Stanford University, Stanford, California, USA
- Department of Bioengineering, Stanford University, Stanford, California, USA
- Institute for Computational and Mathematical Engineering, Stanford University, Stanford, California, USA
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2
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Wülfers EM, Moss R, Lehrmann H, Arentz T, Westermann D, Seemann G, Odening KE, Steinfurt J. Whole-heart computational modelling provides further mechanistic insights into ST-elevation in Brugada syndrome. INTERNATIONAL JOURNAL OF CARDIOLOGY. HEART & VASCULATURE 2024; 51:101373. [PMID: 38464963 PMCID: PMC10924145 DOI: 10.1016/j.ijcha.2024.101373] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/11/2023] [Revised: 02/08/2024] [Accepted: 02/21/2024] [Indexed: 03/12/2024]
Abstract
Background Brugada syndrome (BrS) is characterized by dynamic ST-elevations in right precordial leads and increased risk of ventricular fibrillation and sudden cardiac death. As the mechanism underlying ST-elevation and malignant arrhythmias is controversial computational modeling can aid in exploring the disease mechanism. Thus we aim to test the main competing hypotheses ('delayed depolarization' vs. 'early repolarization') of BrS in a whole-heart computational model. Methods In a 3D whole-heart computational model, delayed epicardial RVOT activation with local conduction delay was simulated by reducing conductivity in the epicardial RVOT. Early repolarization was simulated by instead increasing the transient outward potassium current (Ito) in the same region. Additionally, a reduction in the fast sodium current (INa) was incorporated in both models. Results Delayed depolarization with local conduction delay in the computational model resulted in coved-type ST-elevation with negative T-waves in the precordial surface ECG leads. 'Saddleback'-shaped ST-elevation was obtained with reduced substrate extent or thickness. Increased Ito simulations showed early repolarization in the RVOT with a descending but not coved-type ST-elevation. Reduced INa did not show a significant effect on ECG morphology. Conclusions In this whole-heart BrS computational model of both major hypotheses, realistic coved-type ECG resulted only from delayed epicardial RVOT depolarization with local conduction delay but not early repolarizing ion channel modifications. These simulations provide further support for the depolarization hypothesis as electrophysiological mechanism underlying BrS.
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Affiliation(s)
- Eike M Wülfers
- Institute for Experimental Cardiovascular Medicine, University Heart Center Freiburg - Bad Krozingen, and Faculty of Medicine, University of Freiburg, Freiburg, Germany
- Department of Physics and Astronomy, Faculty of Sciences, Ghent University, Ghent, Belgium
| | - Robin Moss
- Institute for Experimental Cardiovascular Medicine, University Heart Center Freiburg - Bad Krozingen, and Faculty of Medicine, University of Freiburg, Freiburg, Germany
| | - Heiko Lehrmann
- Department of Cardiology and Angiology, University Heart Center Freiburg - Bad Krozingen, and Faculty of Medicine, University of Freiburg, Freiburg, Germany
| | - Thomas Arentz
- Department of Cardiology and Angiology, University Heart Center Freiburg - Bad Krozingen, and Faculty of Medicine, University of Freiburg, Freiburg, Germany
| | - Dirk Westermann
- Department of Cardiology and Angiology, University Heart Center Freiburg - Bad Krozingen, and Faculty of Medicine, University of Freiburg, Freiburg, Germany
| | - Gunnar Seemann
- Institute for Experimental Cardiovascular Medicine, University Heart Center Freiburg - Bad Krozingen, and Faculty of Medicine, University of Freiburg, Freiburg, Germany
| | - Katja E Odening
- Translational Cardiology, Department of Cardiology and Institute of Physiology, University Hospital Bern, University of Bern, Switzerland
| | - Johannes Steinfurt
- Department of Cardiology and Angiology, University Heart Center Freiburg - Bad Krozingen, and Faculty of Medicine, University of Freiburg, Freiburg, Germany
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3
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Salvador M, Marsden AL. Branched Latent Neural Maps. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2024; 418:116499. [PMID: 37872974 PMCID: PMC10588816 DOI: 10.1016/j.cma.2023.116499] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/25/2023]
Abstract
We introduce Branched Latent Neural Maps (BLNMs) to learn finite dimensional input-output maps encoding complex physical processes. A BLNM is defined by a simple and compact feedforward partially-connected neural network that structurally disentangles inputs with different intrinsic roles, such as the time variable from model parameters of a differential equation, while transferring them into a generic field of interest. BLNMs leverage latent outputs to enhance the learned dynamics and break the curse of dimensionality by showing excellent in-distribution generalization properties with small training datasets and short training times on a single processor. Indeed, their in-distribution generalization error remains comparable regardless of the adopted discretization during the testing phase. Moreover, the partial connections, in place of a fully-connected structure, significantly reduce the number of tunable parameters. We show the capabilities of BLNMs in a challenging test case involving biophysically detailed electrophysiology simulations in a biventricular cardiac model of a pediatric patient with hypoplastic left heart syndrome. The model includes a 1D Purkinje network for fast conduction and a 3D heart-torso geometry. Specifically, we trained BLNMs on 150 in silico generated 12-lead electrocardiograms (ECGs) while spanning 7 model parameters, covering cell-scale, organ-level and electrical dyssynchrony. Although the 12-lead ECGs manifest very fast dynamics with sharp gradients, after automatic hyperparameter tuning the optimal BLNM, trained in less than 3 hours on a single CPU, retains just 7 hidden layers and 19 neurons per layer. The resulting mean square error is on the order of 10 - 4 on an independent test dataset comprised of 50 additional electrophysiology simulations. In the online phase, the BLNM allows for 5000x faster real-time simulations of cardiac electrophysiology on a single core standard computer and can be employed to solve inverse problems via global optimization in a few seconds of computational time. This paper provides a novel computational tool to build reliable and efficient reduced-order models for digital twinning in engineering applications. The Julia implementation is publicly available under MIT License at https://github.com/StanfordCBCL/BLNM.jl.
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Affiliation(s)
- Matteo Salvador
- Institute for Computational and Mathematical Engineering, Stanford University, California, USA
- Cardiovascular Institute, Stanford University, California, USA
- Pediatric Cardiology, Stanford University, California, USA
| | - Alison Lesley Marsden
- Department of Bioengineering, Stanford University, California, USA
- Institute for Computational and Mathematical Engineering, Stanford University, California, USA
- Cardiovascular Institute, Stanford University, California, USA
- Pediatric Cardiology, Stanford University, California, USA
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4
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Zingaro A, Vergara C, Dede' L, Regazzoni F, Quarteroni A. A comprehensive mathematical model for cardiac perfusion. Sci Rep 2023; 13:14220. [PMID: 37648701 PMCID: PMC10469210 DOI: 10.1038/s41598-023-41312-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2023] [Accepted: 08/24/2023] [Indexed: 09/01/2023] Open
Abstract
The aim of this paper is to introduce a new mathematical model that simulates myocardial blood perfusion that accounts for multiscale and multiphysics features. Our model incorporates cardiac electrophysiology, active and passive mechanics, hemodynamics, valve modeling, and a multicompartment Darcy model of perfusion. We consider a fully coupled electromechanical model of the left heart that provides input for a fully coupled Navier-Stokes-Darcy Model for myocardial perfusion. The fluid dynamics problem is modeled in a left heart geometry that includes large epicardial coronaries, while the multicompartment Darcy model is set in a biventricular myocardium. Using a realistic and detailed cardiac geometry, our simulations demonstrate the biophysical fidelity of our model in describing cardiac perfusion. Specifically, we successfully validate the model reliability by comparing in-silico coronary flow rates and average myocardial blood flow with clinically established values ranges reported in relevant literature. Additionally, we investigate the impact of a regurgitant aortic valve on myocardial perfusion, and our results indicate a reduction in myocardial perfusion due to blood flow taken away by the left ventricle during diastole. To the best of our knowledge, our work represents the first instance where electromechanics, hemodynamics, and perfusion are integrated into a single computational framework.
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Affiliation(s)
- Alberto Zingaro
- MOX, Laboratory of Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy.
- ELEM Biotech S.L., Pier01, Palau de Mar, Plaça Pau Vila, 1, 08003, Barcelona, Spain.
| | - Christian Vergara
- LaBS, Dipartimento di Chimica, Materiali e Ingegneria Chimica "Giulio Natta", Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy
| | - Luca Dede'
- MOX, Laboratory of Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy
| | - Francesco Regazzoni
- MOX, Laboratory of Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy
| | - Alfio Quarteroni
- MOX, Laboratory of Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy
- Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, Av. Piccard, CH-1015, Lausanne, Switzerland
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5
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Ogiermann D, Perotti LE, Balzani D. A simple and efficient adaptive time stepping technique for low-order operator splitting schemes applied to cardiac electrophysiology. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2023; 39:e3670. [PMID: 36510350 DOI: 10.1002/cnm.3670] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/20/2021] [Revised: 10/25/2022] [Accepted: 12/04/2022] [Indexed: 06/17/2023]
Abstract
We present a simple, yet efficient adaptive time stepping scheme for cardiac electrophysiology (EP) simulations based on standard operator splitting techniques. The general idea is to exploit the relation between the splitting error and the reaction's magnitude-found in a previous one-dimensional analytical study by Spiteri and Ziaratgahi-to construct the new time step controller for three-dimensional problems. Accordingly, we propose to control the time step length of the operator splitting scheme as a function of the reaction magnitude, in addition to the common approach of adapting the reaction time step. This conforms with observations in numerical experiments supporting the need for a significantly smaller time step length during depolarization than during repolarization. The proposed scheme is compared with classical proportional-integral-differential controllers using state-of-the-art error estimators, which are also presented in details as they have not been previously applied in the context of cardiac EP with operator splitting techniques. Benchmarks show that choosing the time step as a sigmoidal function of the reaction magnitude is highly efficient and full cardiac cycles can be computed with precision even in a realistic biventricular setup. The proposed scheme outperforms common adaptive time stepping techniques, while depending on fewer tuning parameters.
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Affiliation(s)
- Dennis Ogiermann
- Chair of Continuum Mechanics, Ruhr University Bochum, Bochum, Germany
| | - Luigi E Perotti
- Mechanical and Aerospace Engineering Department, University of Central Florida, Orlando, Florida, USA
| | - Daniel Balzani
- Chair of Continuum Mechanics, Ruhr University Bochum, Bochum, Germany
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6
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Lindner LP, Gerach T, Jahnke T, Loewe A, Weiss D, Wieners C. Efficient time splitting schemes for the monodomain equation in cardiac electrophysiology. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2023; 39:e3666. [PMID: 36562492 DOI: 10.1002/cnm.3666] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2022] [Revised: 10/19/2022] [Accepted: 11/26/2022] [Indexed: 06/17/2023]
Abstract
Approximating the fast dynamics of depolarization waves in the human heart described by the monodomain model is numerically challenging. Splitting methods for the PDE-ODE coupling enable the computation with very fine space and time discretizations. Here, we compare different splitting approaches regarding convergence, accuracy, and efficiency. Simulations were performed for a benchmark problem with the Beeler-Reuter cell model on a truncated ellipsoid approximating the left ventricle including a localized stimulation. For this configuration, we provide a reference solution for the transmembrane potential. We found a semi-implicit approach with state variable interpolation to be the most efficient scheme. The results are transferred to a more physiological setup using a bi-ventricular domain with a complex external stimulation pattern to evaluate the accuracy of the activation time for different resolutions in space and time.
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Affiliation(s)
- Laura P Lindner
- Institute of Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
| | - Tobias Gerach
- Institute of Biomedical Engineering, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
| | - Tobias Jahnke
- Institute of Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
| | - Axel Loewe
- Institute of Biomedical Engineering, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
| | - Daniel Weiss
- Institute of Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
| | - Christian Wieners
- Institute of Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
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7
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Rossi S, Abdala L, Woodward A, Vavalle JP, Henriquez CS, Griffith BE. Rule-based definition of muscle bundles in patient-specific models of the left atrium. Front Physiol 2022; 13:912947. [PMID: 36311246 PMCID: PMC9597256 DOI: 10.3389/fphys.2022.912947] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2022] [Accepted: 06/29/2022] [Indexed: 11/13/2022] Open
Abstract
Atrial fibrillation (AF) is the most common arrhythmia encountered clinically, and as the population ages, its prevalence is increasing. Although the CHA2DS2−VASc score is the most used risk-stratification system for stroke risk in AF, it lacks personalization. Patient-specific computer models of the atria can facilitate personalized risk assessment and treatment planning. However, a challenge faced in creating such models is the complexity of the atrial muscle arrangement and its influence on the atrial fiber architecture. This work proposes a semi-automated rule-based algorithm to generate the local fiber orientation in the left atrium (LA). We use the solutions of several harmonic equations to decompose the LA anatomy into subregions. Solution gradients define a two-layer fiber field in each subregion. The robustness of our approach is demonstrated by recreating the fiber orientation on nine models of the LA obtained from AF patients who underwent WATCHMAN device implantation. This cohort of patients encompasses a variety of morphology variants of the left atrium, both in terms of the left atrial appendages (LAAs) and the number of pulmonary veins (PVs). We test the fiber construction algorithm by performing electrophysiology (EP) simulations. Furthermore, this study is the first to compare its results with other rule-based algorithms for the LA fiber architecture definition available in the literature. This analysis suggests that a multi-layer fiber architecture is important to capture complex electrical activation patterns. A notable advantage of our approach is the ability to reconstruct the main LA fiber bundles in a variety of morphologies while solving for a small number of harmonic fields, leading to a comparatively straightforward and reproducible approach.
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Affiliation(s)
- Simone Rossi
- Department of Mathematics, UNC Chapel Hill, Chapel Hill, NC, United States
- *Correspondence: Simone Rossi, ; Boyce E. Griffith,
| | - Laryssa Abdala
- Department of Mathematics, UNC Chapel Hill, Chapel Hill, NC, United States
| | - Andrew Woodward
- Advanced Medical Imaging Lab, UNC Chapel Hill, Chapel Hill, NC, United States
| | - John P. Vavalle
- Department of Medicine, UNC Chapel Hill, Chapel Hill, NC, United States
| | - Craig S. Henriquez
- Department of Biomedical Engineering, Duke University, Durham, NC, United States
| | - Boyce E. Griffith
- Department of Mathematics, UNC Chapel Hill, Chapel Hill, NC, United States
- Department of Biomedical Engineering, UNC Chapel Hill, Chapel Hill, NC, United States
- McAllister Heart Institute, UNC Chapel Hill, Chapel Hill, NC, United States
- *Correspondence: Simone Rossi, ; Boyce E. Griffith,
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8
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Woodworth LA, Cansız B, Kaliske M. Balancing conduction velocity error in cardiac electrophysiology using a modified quadrature approach. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2022; 38:e3589. [PMID: 35266643 DOI: 10.1002/cnm.3589] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2021] [Revised: 01/20/2022] [Accepted: 02/28/2022] [Indexed: 06/14/2023]
Abstract
Conduction velocity error is often the main culprit behind the need for very fine spatial discretizations and high computational effort in cardiac electrophysiology problems. In light of this, a novel approach for simulating an accurate conduction velocity in coarse meshes with linear elements is suggested based on a modified quadrature approach. In this approach, the quadrature points are placed at arbitrary offsets of the isoparametric coordinates. A numerical study illustrates the dependence of the conduction velocity on the spatial discretization and the conductivity when using different quadrature rules and calculation approaches. Additionally, examples using the modified quadrature in coarse meshes for wave propagation demonstrate the improved accuracy of the conduction velocity with this method. This novel approach possesses great potential in reducing the computational effort required but remains limited to specific linear elements and experiences a reduction in accuracy for irregular meshes and heterogeneous conductivities. Further research can focus on developing an adaptive quadrature and extending the approach to other element formulations in order to make the approach more generally applicable.
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Affiliation(s)
- Lucas A Woodworth
- Institute for Structural Analysis, Technische Universität Dresden, Dresden, Germany
| | - Barış Cansız
- Institute for Structural Analysis, Technische Universität Dresden, Dresden, Germany
| | - Michael Kaliske
- Institute for Structural Analysis, Technische Universität Dresden, Dresden, Germany
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9
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Gander L, Pezzuto S, Gharaviri A, Krause R, Perdikaris P, Sahli Costabal F. Fast Characterization of Inducible Regions of Atrial Fibrillation Models With Multi-Fidelity Gaussian Process Classification. Front Physiol 2022; 13:757159. [PMID: 35330935 PMCID: PMC8940533 DOI: 10.3389/fphys.2022.757159] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/11/2021] [Accepted: 01/31/2022] [Indexed: 11/13/2022] Open
Abstract
Computational models of atrial fibrillation have successfully been used to predict optimal ablation sites. A critical step to assess the effect of an ablation pattern is to pace the model from different, potentially random, locations to determine whether arrhythmias can be induced in the atria. In this work, we propose to use multi-fidelity Gaussian process classification on Riemannian manifolds to efficiently determine the regions in the atria where arrhythmias are inducible. We build a probabilistic classifier that operates directly on the atrial surface. We take advantage of lower resolution models to explore the atrial surface and combine seamlessly with high-resolution models to identify regions of inducibility. We test our methodology in 9 different cases, with different levels of fibrosis and ablation treatments, totalling 1,800 high resolution and 900 low resolution simulations of atrial fibrillation. When trained with 40 samples, our multi-fidelity classifier that combines low and high resolution models, shows a balanced accuracy that is, on average, 5.7% higher than a nearest neighbor classifier. We hope that this new technique will allow faster and more precise clinical applications of computational models for atrial fibrillation. All data and code accompanying this manuscript will be made publicly available at: https://github.com/fsahli/AtrialMFclass.
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Affiliation(s)
- Lia Gander
- Center for Computational Medicine in Cardiology, Euler Institute, Università della Svizzera italiana, Lugano, Switzerland
| | - Simone Pezzuto
- Center for Computational Medicine in Cardiology, Euler Institute, Università della Svizzera italiana, Lugano, Switzerland
| | - Ali Gharaviri
- Center for Computational Medicine in Cardiology, Euler Institute, Università della Svizzera italiana, Lugano, Switzerland
| | - Rolf Krause
- Center for Computational Medicine in Cardiology, Euler Institute, Università della Svizzera italiana, Lugano, Switzerland
| | - Paris Perdikaris
- Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, United States
| | - Francisco Sahli Costabal
- Department of Mechanical and Metallurgical Engineering, School of Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile.,Institute for Biological and Medical Engineering, Schools of Engineering, Medicine and Biological Sciences, Pontificia Universidad Católica de Chile, Santiago, Chile.,Millennium Nucleus for Cardiovascular Magnetic Resonance, Santiago, Chile
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10
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Woodworth LA, Cansız B, Kaliske M. A numerical study on the effects of spatial and temporal discretization in cardiac electrophysiology. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2021; 37:e3443. [PMID: 33522111 DOI: 10.1002/cnm.3443] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2020] [Revised: 12/16/2020] [Accepted: 01/23/2021] [Indexed: 06/12/2023]
Abstract
Millions of degrees of freedom are often required to accurately represent the electrophysiology of the myocardium due to the presence of discretization effects. This study seeks to explore the influence of temporal and spatial discretization on the simulation of cardiac electrophysiology in conjunction with changes in modeling choices. Several finite element analyses are performed to examine how discretization affects solution time, conduction velocity and electrical excitation. Discretization effects are considered along with changes in the electrophysiology model and solution approach. Two action potential models are considered: the Aliev-Panfilov model and the ten Tusscher-Noble-Noble-Panfilov model. The solution approaches consist of two time integration schemes and different treatments for solving the local system of ordinary differential equations. The efficiency and stability of the calculation approaches are demonstrated to be dependent on the action potential model. The dependency of the conduction velocity on the element size and time step is shown to be different for changes in material parameters. Finally, the discrepancies between the wave propagation in coarse and fine meshes are analyzed based on the temporal evolution of the transmembrane potential at a node and its neighboring Gauss points. Insight obtained from this study can be used to suggest new methods to improve the efficiency of simulations in cardiac electrophysiology.
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Affiliation(s)
- Lucas A Woodworth
- Institute for Structural Analysis, Technische Universität Dresden, Dresden, Germany
| | - Barış Cansız
- Institute for Structural Analysis, Technische Universität Dresden, Dresden, Germany
| | - Michael Kaliske
- Institute for Structural Analysis, Technische Universität Dresden, Dresden, Germany
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11
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Clayton RH, Aboelkassem Y, Cantwell CD, Corrado C, Delhaas T, Huberts W, Lei CL, Ni H, Panfilov AV, Roney C, dos Santos RW. An audit of uncertainty in multi-scale cardiac electrophysiology models. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190335. [PMID: 32448070 PMCID: PMC7287340 DOI: 10.1098/rsta.2019.0335] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 03/16/2020] [Indexed: 05/21/2023]
Abstract
Models of electrical activation and recovery in cardiac cells and tissue have become valuable research tools, and are beginning to be used in safety-critical applications including guidance for clinical procedures and for drug safety assessment. As a consequence, there is an urgent need for a more detailed and quantitative understanding of the ways that uncertainty and variability influence model predictions. In this paper, we review the sources of uncertainty in these models at different spatial scales, discuss how uncertainties are communicated across scales, and begin to assess their relative importance. We conclude by highlighting important challenges that continue to face the cardiac modelling community, identifying open questions, and making recommendations for future studies. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.
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Affiliation(s)
- Richard H. Clayton
- Insigneo institute for in-silico Medicine and Department of Computer Science, University of Sheffield, Sheffield, UK
- e-mail:
| | - Yasser Aboelkassem
- Department of Bioengineering, University of California, San Diego, CA, USA
| | | | - Cesare Corrado
- Division of Imaging Sciences and Biomedical Engineering, King’s College London, London, UK
| | - Tammo Delhaas
- School of Cardiovascular Diseases, Maastricht University, Maastricht, The Netherlands
| | - Wouter Huberts
- School of Cardiovascular Diseases, Maastricht University, Maastricht, The Netherlands
- Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
| | - Chon Lok Lei
- Computational Biology and Health Informatics, Department of Computer Science, University of Oxford, Oxford, UK
| | - Haibo Ni
- Department of Pharmacology, University of California, Davis, CA, USA
| | - Alexander V. Panfilov
- Department of Physics and Astronomy, University of Gent, Gent, Belgium
- Laboratory of Computational Biology and Medicine, Ural Federal University, Ekaterinburg, Russia
| | - Caroline Roney
- Division of Imaging Sciences and Biomedical Engineering, King’s College London, London, UK
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12
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Hurtado DE, Jilberto J, Panasenko G. Non-ohmic tissue conduction in cardiac electrophysiology: Upscaling the non-linear voltage-dependent conductance of gap junctions. PLoS Comput Biol 2020; 16:e1007232. [PMID: 32097410 PMCID: PMC7059938 DOI: 10.1371/journal.pcbi.1007232] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2019] [Revised: 03/06/2020] [Accepted: 01/15/2020] [Indexed: 12/21/2022] Open
Abstract
Gap junctions are key mediators of intercellular communication in cardiac tissue, and their function is vital to sustaining normal cardiac electrical activity. Conduction through gap junctions strongly depends on the hemichannel arrangement and transjunctional voltage, rendering the intercellular conductance highly non-Ohmic, particularly under steady-state regimes of conduction. Despite this marked non-linear behavior, current tissue-level models of cardiac conduction are rooted in the assumption that gap-junctions conductance is constant (Ohmic), which results in inaccurate predictions of electrical propagation, particularly in the low junctional-coupling regime observed under pathological conditions. In this work, we present a novel non-Ohmic homogenization model (NOHM) of cardiac conduction that is suitable to tissue-scale simulations. Using non-linear homogenization theory, we develop a conductivity model that seamlessly upscales the voltage-dependent conductance of gap junctions, without the need of explicitly modeling gap junctions. The NOHM model allows for the simulation of electrical propagation in tissue-level cardiac domains that accurately resemble that of cell-based microscopic models for a wide range of junctional coupling scenarios, recovering key conduction features at a fraction of the computational complexity. A unique feature of the NOHM model is the possibility of upscaling the response of non-symmetric gap-junction conductance distributions, which result in conduction velocities that strongly depend on the direction of propagation, thus allowing to model the normal and retrograde conduction observed in certain regions of the heart. We envision that the NOHM model will enable organ-level simulations that are informed by sub- and inter-cellular mechanisms, delivering an accurate and predictive in-silico tool for understanding the heart function. Codes are available for download at https://github.com/dehurtado/NonOhmicConduction. The heart relies on the propagation of electrical impulses that are mediated gap junctions, whose conduction properties vary depending on the transjunctional voltage. Despite this non-linear feature, current mathematical models assume that cardiac tissue behaves like an Ohmic (linear) material, thus delivering inaccurate results when simulated in a computer. Here we present a novel mathematical multiscale model that explicitly includes the non-Ohmic response of gap junctions in its predictions. Our results show that the proposed model recovers important conduction features modulated by gap junctions at a fraction of the computational complexity. This contribution represents an important step towards constructing computer models of a whole heart that can predict organ-level behavior in reasonable computing times.
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Affiliation(s)
- Daniel E. Hurtado
- Institute for Biological and Medical Engineering, Schools of Engineering, Medicine and Biological Sciences, Pontificia Universidad Católica de Chile, Santiago, Chile
- Department of Structural and Geotechnical Engineering, School of Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile
- Millennium Nucleus for Cardiovascular Magnetic Resonance, Chile
- * E-mail:
| | - Javiera Jilberto
- Institute for Biological and Medical Engineering, Schools of Engineering, Medicine and Biological Sciences, Pontificia Universidad Católica de Chile, Santiago, Chile
- Millennium Nucleus for Cardiovascular Magnetic Resonance, Chile
| | - Grigory Panasenko
- Institute Camille Jordan, Université Jean Monnet, Université de Lyon, Saint-Etienne, France
- Institute of Applied Mathematics, Vilnius University, Vilnius, Lithuania
- National Research University “Moscow Power Engineering Institute”, Moscow, Russia
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13
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Pezzuto S, Gharaviri A, Schotten U, Potse M, Conte G, Caputo ML, Regoli F, Krause R, Auricchio A. Beat-to-beat P-wave morphological variability in patients with paroxysmal atrial fibrillation: anin silicostudy. Europace 2018; 20:iii26-iii35. [DOI: 10.1093/europace/euy227] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2018] [Accepted: 09/19/2018] [Indexed: 11/12/2022] Open
Affiliation(s)
- Simone Pezzuto
- Center for Computational Medicine in Cardiology, Università della Svizzera italiana, Lugano, Switzerland
- Institute of Computational Science, Università della Svizzera italiana, Via Giuseppe Buffi 13, CH-6904 Lugano, Switzerland
| | - Ali Gharaviri
- Center for Computational Medicine in Cardiology, Università della Svizzera italiana, Lugano, Switzerland
- Institute of Computational Science, Università della Svizzera italiana, Via Giuseppe Buffi 13, CH-6904 Lugano, Switzerland
| | - Ulrich Schotten
- Department of Physiology, Cardiovascular Research Institute Maastricht, Maastricht University, Maastricht, the Netherlands
| | - Mark Potse
- CARMEN Research Team, INRIA, Talence, France
- IHU Liryc, Electrophysiology and Heart Modeling Institute, Foundation Bordeaux Université, Pessac, France
- Univ. Bordeaux, IMB, UMR 5251, Talence, France
| | - Giulio Conte
- Center for Computational Medicine in Cardiology, Università della Svizzera italiana, Lugano, Switzerland
- Fondazione Cardiocentro Ticino, Lugano, Switzerland
| | - Maria Luce Caputo
- Center for Computational Medicine in Cardiology, Università della Svizzera italiana, Lugano, Switzerland
- Fondazione Cardiocentro Ticino, Lugano, Switzerland
| | - Francois Regoli
- Center for Computational Medicine in Cardiology, Università della Svizzera italiana, Lugano, Switzerland
- Fondazione Cardiocentro Ticino, Lugano, Switzerland
| | - Rolf Krause
- Center for Computational Medicine in Cardiology, Università della Svizzera italiana, Lugano, Switzerland
- Institute of Computational Science, Università della Svizzera italiana, Via Giuseppe Buffi 13, CH-6904 Lugano, Switzerland
| | - Angelo Auricchio
- Center for Computational Medicine in Cardiology, Università della Svizzera italiana, Lugano, Switzerland
- Fondazione Cardiocentro Ticino, Lugano, Switzerland
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14
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Jilberto J, Hurtado DE. Semi-implicit Non-conforming Finite-Element Schemes for Cardiac Electrophysiology: A Framework for Mesh-Coarsening Heart Simulations. Front Physiol 2018; 9:1513. [PMID: 30425648 PMCID: PMC6218665 DOI: 10.3389/fphys.2018.01513] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2018] [Accepted: 10/09/2018] [Indexed: 11/13/2022] Open
Abstract
The field of computational cardiology has steadily progressed toward reliable and accurate simulations of the heart, showing great potential in clinical applications such as the optimization of cardiac interventions and the study of pro-arrhythmic effects of drugs in humans, among others. However, the computational effort demanded by in-silico studies of the heart remains challenging, highlighting the need of novel numerical methods that can improve the efficiency of simulations while targeting an acceptable accuracy. In this work, we propose a semi-implicit non-conforming finite-element scheme (SINCFES) suitable for cardiac electrophysiology simulations. The accuracy and efficiency of the proposed scheme are assessed by means of numerical simulations of the electrical excitation and propagation in regular and biventricular geometries. We show that the SINCFES allows for coarse-mesh simulations that reduce the computation time when compared to fine-mesh models while delivering wavefront shapes and conduction velocities that are more accurate than those predicted by traditional finite-element formulations based on the same coarse mesh, thus improving the accuracy-efficiency trade-off of cardiac simulations.
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Affiliation(s)
- Javiera Jilberto
- Department of Structural and Geotechnical Engineering, School of Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile
| | - Daniel E. Hurtado
- Department of Structural and Geotechnical Engineering, School of Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile
- Institute for Biological and Medical Engineering, Schools of Engineering, Medicine and Biological Sciences, Pontificia Universidad Católica de Chile, Santiago, Chile
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15
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Quaglino A, Pezzuto S, Koutsourelakis PS, Auricchio A, Krause R. Fast uncertainty quantification of activation sequences in patient-specific cardiac electrophysiology meeting clinical time constraints. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2018; 34:e2985. [PMID: 29577657 DOI: 10.1002/cnm.2985] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2017] [Revised: 01/16/2018] [Accepted: 03/15/2018] [Indexed: 06/08/2023]
Abstract
We present a fast, patient-specific methodology for uncertainty quantification in electrophysiology, aimed at meeting the time constraints of clinical practitioners. We focus on computing the statistics of the activation map, given the uncertainties associated with the conductivity tensor modeling the fiber orientation in the heart. We use a fast parallel solution method implemented on a graphics processing unit for the eikonal approximation, in order to compute the activation map and to sample the random fiber field with correlation on the basis of geodesic distances. While this enables to perform uncertainty quantification studies with a manageable computational effort, the required time frame still exceeds clinically suitable time expectations. In order to reduce it further by 2 orders of magnitude, we rely on Bayesian multifidelity methods. In particular, we propose a low-fidelity model that is patient-specific and free from the additional training cost associated with reduced models. This is achieved by a sound physics-based simplification of the full eikonal model. The low-fidelity output is then corrected by the standard multifidelity framework. In practice, the complete procedure only requires approximately 100 new runs of our eikonal graphics processing unit solver for producing the sought estimates and their associated credible intervals, enabling a full online analysis in less than 5 minutes.
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Affiliation(s)
- A Quaglino
- Center for Computational Medicine in Cardiology, Institute of Computational Science, Università della Svizzera italiana, Lugano, Switzerland
| | - S Pezzuto
- Center for Computational Medicine in Cardiology, Institute of Computational Science, Università della Svizzera italiana, Lugano, Switzerland
| | | | - A Auricchio
- Center for Computational Medicine in Cardiology, Institute of Computational Science, Università della Svizzera italiana, Lugano, Switzerland
- Division of Cardiology, Fondazione Cardiocentro Ticino, Lugano, Switzerland
| | - R Krause
- Center for Computational Medicine in Cardiology, Institute of Computational Science, Università della Svizzera italiana, Lugano, Switzerland
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16
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Hoermann JM, Bertoglio C, Kronbichler M, Pfaller MR, Chabiniok R, Wall WA. An adaptive hybridizable discontinuous Galerkin approach for cardiac electrophysiology. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2018; 34:e2959. [PMID: 29316340 DOI: 10.1002/cnm.2959] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2017] [Revised: 12/13/2017] [Accepted: 12/27/2017] [Indexed: 06/07/2023]
Abstract
Cardiac electrophysiology simulations are numerically challenging because of the propagation of a steep electrochemical wave front and thus require discretizations with small mesh sizes to obtain accurate results. In this work, we present an approach based on the hybridizable discontinuous Galerkin method (HDG), which allows an efficient implementation of high-order discretizations into a computational framework. In particular, using the advantage of the discontinuous function space, we present an efficient p-adaptive strategy for accurately tracking the wave front. The HDG allows to reduce the overall degrees of freedom in the final linear system to those only on the element interfaces. Additionally, we propose a rule for a suitable integration accuracy for the ionic current term depending on the polynomial order and the cell model to handle high-order polynomials. Our results show that for the same number of degrees of freedom, coarse high-order elements provide more accurate results than fine low-order elements. Introducing p-adaptivity further reduces computational costs while maintaining accuracy by restricting the use of high-order elements to resolve the wave front. For a patient-specific simulation of a cardiac cycle, p-adaptivity reduces the average number of degrees of freedom by 95% compared to the nonadaptive model. In addition to reducing computational costs, using coarse meshes with our p-adaptive high-order HDG method also simplifies practical aspects of mesh generation and postprocessing.
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Affiliation(s)
- Julia M Hoermann
- Institute for Computational Mechanics, Technical University Munich, Boltzmannstr 15, Garching b. München, 85748, Germany
| | - Cristóbal Bertoglio
- Center for Mathematical Modeling, Universidad de Chile, Beaucheff 851, Santiago 8370456, Chile
- Johann Bernoulli Institute, University of Groningen, Nijenborgh 9, Groningen, 9747 HZ, Netherlands
| | - Martin Kronbichler
- Institute for Computational Mechanics, Technical University Munich, Boltzmannstr 15, Garching b. München, 85748, Germany
| | - Martin R Pfaller
- Institute for Computational Mechanics, Technical University Munich, Boltzmannstr 15, Garching b. München, 85748, Germany
| | - Radomir Chabiniok
- Inria, Paris-Saclay University, Palaiseau, France
- LMS, Ecole Polytechnique, CNRS, Paris-Saclay University, Palaiseau, France
- School of Biomedical Engineering and Imaging Sciences (BMEIS), St Thomas' Hospital, King's College, London, UK
| | - Wolfgang A Wall
- Institute for Computational Mechanics, Technical University Munich, Boltzmannstr 15, Garching b. München, 85748, Germany
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17
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Rossi S, Griffith BE. Incorporating inductances in tissue-scale models of cardiac electrophysiology. CHAOS (WOODBURY, N.Y.) 2017; 27:093926. [PMID: 28964127 PMCID: PMC5585078 DOI: 10.1063/1.5000706] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2017] [Accepted: 07/17/2017] [Indexed: 06/07/2023]
Abstract
In standard models of cardiac electrophysiology, including the bidomain and monodomain models, local perturbations can propagate at infinite speed. We address this unrealistic property by developing a hyperbolic bidomain model that is based on a generalization of Ohm's law with a Cattaneo-type model for the fluxes. Further, we obtain a hyperbolic monodomain model in the case that the intracellular and extracellular conductivity tensors have the same anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is equivalent to a cable model that includes axial inductances, and the relaxation times of the Cattaneo fluxes are strictly related to these inductances. A purely linear analysis shows that the inductances are negligible, but models of cardiac electrophysiology are highly nonlinear, and linear predictions may not capture the fully nonlinear dynamics. In fact, contrary to the linear analysis, we show that for simple nonlinear ionic models, an increase in conduction velocity is obtained for small and moderate values of the relaxation time. A similar behavior is also demonstrated with biophysically detailed ionic models. Using the Fenton-Karma model along with a low-order finite element spatial discretization, we numerically analyze differences between the standard monodomain model and the hyperbolic monodomain model. In a simple benchmark test, we show that the propagation of the action potential is strongly influenced by the alignment of the fibers with respect to the mesh in both the parabolic and hyperbolic models when using relatively coarse spatial discretizations. Accurate predictions of the conduction velocity require computational mesh spacings on the order of a single cardiac cell. We also compare the two formulations in the case of spiral break up and atrial fibrillation in an anatomically detailed model of the left atrium, and we examine the effect of intracellular and extracellular inductances on the virtual electrode phenomenon.
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Affiliation(s)
- Simone Rossi
- Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA
| | - Boyce E Griffith
- Departments of Mathematics and Biomedical Engineering and McAllister Heart Institute, University of North Carolina, Chapel Hill, North Carolina 27599, USA
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18
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Pezzuto S, Kal'avský P, Potse M, Prinzen FW, Auricchio A, Krause R. Evaluation of a Rapid Anisotropic Model for ECG Simulation. Front Physiol 2017; 8:265. [PMID: 28512434 PMCID: PMC5411438 DOI: 10.3389/fphys.2017.00265] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2016] [Accepted: 04/11/2017] [Indexed: 11/29/2022] Open
Abstract
State-of-the-art cardiac electrophysiology models that are able to deliver physiologically motivated activation maps and electrocardiograms (ECGs) can only be solved on high-performance computing architectures. This makes it nearly impossible to adopt such models in clinical practice. ECG imaging tools typically rely on simplified models, but these neglect the anisotropic electric conductivity of the tissue in the forward problem. Moreover, their results are often confined to the heart-torso interface. We propose a forward model that fully accounts for the anisotropic tissue conductivity and produces the standard 12-lead ECG in a few seconds. The activation sequence is approximated with an eikonal model in the 3d myocardium, while the ECG is computed with the lead-field approach. Both solvers were implemented on graphics processing units and massively parallelized. We studied the numerical convergence and scalability of the approach. We also compared the method to the bidomain model in terms of ECGs and activation maps, using a simplified but physiologically motivated geometry and 6 patient-specific anatomies. The proposed methods provided a good approximation of activation maps and ECGs computed with a bidomain model, in only a few seconds. Both solvers scaled very well to high-end hardware. These methods are suitable for use in ECG imaging methods, and may soon become fast enough for use in interactive simulation tools.
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Affiliation(s)
- Simone Pezzuto
- Center for Computational Medicine in CardiologyLugano, Switzerland
- Faculty of Informatics, Institute of Computational Science, Università della Svizzera ItalianaLugano, Switzerland
| | - Peter Kal'avský
- Center for Computational Medicine in CardiologyLugano, Switzerland
- Department of Biomeasurements, Institute of Measurement Science, Slovak Academy of SciencesBratislava, Slovakia
| | - Mark Potse
- Center for Computational Medicine in CardiologyLugano, Switzerland
- Electrophysiology and Heart Modeling Institute IHU LIRYCPessac, France
- Inria Bordeaux Sud-OuestTalence, France
| | - Frits W. Prinzen
- Center for Computational Medicine in CardiologyLugano, Switzerland
- Department of Physiology, Cardiovascular Research Institute Maastricht, Maastricht UniversityMaastricht, Netherlands
| | - Angelo Auricchio
- Center for Computational Medicine in CardiologyLugano, Switzerland
- Division of Cardiology, Fondazione Cardiocentro TicinoLugano, Switzerland
| | - Rolf Krause
- Center for Computational Medicine in CardiologyLugano, Switzerland
- Faculty of Informatics, Institute of Computational Science, Università della Svizzera ItalianaLugano, Switzerland
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19
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Parametrization of activation based cardiac electrophysiology models using bidomain model simulations. CURRENT DIRECTIONS IN BIOMEDICAL ENGINEERING 2016. [DOI: 10.1515/cdbme-2016-0135] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Abstract
Eikonal models are useful to compute approximate solutions of cardiac excitation propagation in a computationally efficient way. In this work the underlying conduction velocities for different cell types were computed solving the classical bidomain model equations for planar wavefront propagation. It was further investigated how changes in the conductivity tensors within the bidomain model analytically correspond to changes in the conduction velocity. The error in the presence of local front curvature for the derived eikonal model parametrization were analyzed. The conduction velocity simulated based on the bidomain model was overestimated by a maximum of 10%.
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