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Farquhar ME, Burrage K, Weber Dos Santos R, Bueno-Orovio A, Lawson BA. Graph-based homogenisation for modelling cardiac fibrosis. JOURNAL OF COMPUTATIONAL PHYSICS 2022; 459:None. [PMID: 35959500 PMCID: PMC9352598 DOI: 10.1016/j.jcp.2022.111126] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2021] [Revised: 02/15/2022] [Accepted: 03/02/2022] [Indexed: 05/02/2023]
Abstract
Fibrosis, the excess of extracellular matrix, can affect, and even block, propagation of action potential in cardiac tissue. This can result in deleterious effects on heart function, but the nature and severity of these effects depend strongly on the localisation of fibrosis and its by-products in cardiac tissue, such as collagen scar formation. Computer simulation is an important means of understanding the complex effects of fibrosis on activation patterns in the heart, but concerns of computational cost place restrictions on the spatial resolution of these simulations. In this work, we present a novel numerical homogenisation technique that uses both Eikonal and graph approaches to allow fine-scale heterogeneities in conductivity to be incorporated into a coarser mesh. Homogenisation achieves this by deriving effective conductivity tensors so that a coarser mesh can then be used for numerical simulation. By taking a graph-based approach, our homogenisation technique functions naturally on irregular grids and does not rely upon any assumptions of periodicity, even implicitly. We present results of action potential propagation through fibrotic tissue in two dimensions that show the graph-based homogenisation technique is an accurate and effective way to capture fine-scale domain information on coarser meshes in the context of sharp-fronted travelling waves of activation. As test problems, we consider excitation propagation in tissue with diffuse fibrosis and through a tunnel-like structure designed to test homogenisation, interaction of an excitation wave with a scar region, and functional re-entry.
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Affiliation(s)
- Megan E. Farquhar
- Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers, School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Kevin Burrage
- Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers, School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- Department of Computer Science, Oxford University, Oxford, United Kingdom
| | - Rodrigo Weber Dos Santos
- Department of Computer Science and Program on Computational Modeling, Universidade Federal de Juiz de Fora, Juiz de Fora, Brazil
| | | | - Brodie A.J. Lawson
- Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers, School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
- Centre for Data Science, Queensland University of Technology, Brisbane, Australia
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2
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Irakoze É, Jacquemet V. Multiparameter optimization of nonuniform passive diffusion properties for creating coarse-grained equivalent models of cardiac propagation. Comput Biol Med 2021; 138:104863. [PMID: 34562679 DOI: 10.1016/j.compbiomed.2021.104863] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2021] [Revised: 09/06/2021] [Accepted: 09/07/2021] [Indexed: 11/30/2022]
Abstract
The arrhythmogenic role of discrete cardiac propagation may be assessed by comparing discrete (fine-grained) and equivalent continuous (coarse-grained) models. We aim to develop an optimization algorithm for estimating the smooth conductivity field that best reproduces the diffusion properties of a given discrete model. Our algorithm iteratively adjusts local conductivity of the coarse-grained continuous model by simulating passive diffusion from white noise initial conditions during 3-10 ms and computing the root mean square error with respect to the discrete model. The coarse-grained conductivity field was interpolated from up to 300 evenly spaced control points. We derived an approximate formula for the gradient of the cost function that required (in two dimensions) only two additional simulations per iteration regardless of the number of estimated parameters. Conjugate gradient solver facilitated simultaneous optimization of multiple conductivity parameters. The method was tested in rectangular anisotropic tissues with uniform and nonuniform conductivity (slow regions with sinusoidal profile) and random diffuse fibrosis, as well as in a monolayer interconnected cable model of the left atrium with spatially-varying fibrosis density. Comparison of activation maps served as validation. The results showed that after convergence the errors in activation time were < 1 ms for rectangular geometries and 1-3 ms in the atrial model. Our approach based on the comparison of passive properties (<10 ms simulation) avoids performing active propagation simulations (>100 ms) at each iteration while reproducing activation maps, with possible applications to investigating the impact of microstructure on arrhythmias.
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Affiliation(s)
- Éric Irakoze
- Pharmacology and Physiology Department, Institute of Biomedical Engineering, Université de Montréal, Montreal, QC, H3T 1J4, Canada; Hôpital Du Sacré-Cœur de Montréal, Research Center, 5400 Boul. Gouin Ouest, Montreal, QC, H4J 1C5, Canada
| | - Vincent Jacquemet
- Pharmacology and Physiology Department, Institute of Biomedical Engineering, Université de Montréal, Montreal, QC, H3T 1J4, Canada; Hôpital Du Sacré-Cœur de Montréal, Research Center, 5400 Boul. Gouin Ouest, Montreal, QC, H4J 1C5, Canada.
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3
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Sun M, de Groot NMS, Hendriks RC. Cardiac tissue conductivity estimation using confirmatory factor analysis. Comput Biol Med 2021; 135:104604. [PMID: 34217979 DOI: 10.1016/j.compbiomed.2021.104604] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2021] [Revised: 06/05/2021] [Accepted: 06/21/2021] [Indexed: 11/19/2022]
Abstract
Impaired electrical conduction has been shown to play an important role in the development of heart rhythm disorders. Being able to determine the conductivity is important to localize the arrhythmogenic substrate that causes abnormalities in atrial tissue. In this work, we present an algorithm to estimate the conductivity from epicardial electrograms (EGMs) using a high-resolution electrode array. With these arrays, it is possible to measure the propagation of the extracellular potential of the cardiac tissue at multiple positions simultaneously. Given this data, it is in principle possible to estimate the tissue conductivity. However, this is an ill-posed problem due to the large number of unknown parameters in the electrophysiological data model. In this paper, we make use of an effective method called confirmatory factor analysis (CFA), which we apply to the cross correlation matrix of the data to estimate the tissue conductivity. CFA comes with identifiability conditions that need to be satisfied to solve the problem, which is, in this case, estimation of the tissue conductivity. These identifiability conditions can be used to find the relationship between the desired resolution and the required amount of data. Numerical experiments on the simulated data demonstrate that the proposed method can localize the conduction blocks in the tissue and can also estimate the smoother variation in the conductivities. The conductivity values estimated from the clinical data are in line with the values reported in literature and the EGMs reconstructed based on the estimated parameters match well with the clinical EGMs.
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Affiliation(s)
- Miao Sun
- Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, the Netherlands.
| | | | - Richard C Hendriks
- Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, the Netherlands
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Approaches for determining cardiac bidomain conductivity values: progress and challenges. Med Biol Eng Comput 2020; 58:2919-2935. [PMID: 33089458 DOI: 10.1007/s11517-020-02272-z] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2020] [Accepted: 09/17/2020] [Indexed: 10/23/2022]
Abstract
Modelling the electrical activity of the heart is an important tool for understanding electrical function in various diseases and conduction disorders. Clearly, for model results to be useful, it is necessary to have accurate inputs for the models, in particular the commonly used bidomain model. However, there are only three sets of four experimentally determined conductivity values for cardiac ventricular tissue and these are inconsistent, were measured around 40 years ago, often produce different results in simulations and do not fully represent the three-dimensional anisotropic nature of cardiac tissue. Despite efforts in the intervening years, difficulties associated with making the measurements and also determining the conductivities from the experimental data have not yet been overcome. In this review, we summarise what is known about the conductivity values, as well as progress to date in meeting the challenges associated with both the mathematical modelling and the experimental techniques. Graphical abstract Epicardial potential distributions, arising from a subendocardial ischaemic region, modelled using conductivity data from the indicated studies.
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Hoffman MJ, Cherry EM. Sensitivity of a data-assimilation system for reconstructing three-dimensional cardiac electrical dynamics. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190388. [PMID: 32448069 PMCID: PMC7287341 DOI: 10.1098/rsta.2019.0388] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 02/25/2020] [Indexed: 05/21/2023]
Abstract
Modelling of cardiac electrical behaviour has led to important mechanistic insights, but important challenges, including uncertainty in model formulations and parameter values, make it difficult to obtain quantitatively accurate results. An alternative approach is combining models with observations from experiments to produce a data-informed reconstruction of system states over time. Here, we extend our earlier data-assimilation studies using an ensemble Kalman filter to reconstruct a three-dimensional time series of states with complex spatio-temporal dynamics using only surface observations of voltage. We consider the effects of several algorithmic and model parameters on the accuracy of reconstructions of known scroll-wave truth states using synthetic observations. In particular, we study the algorithm's sensitivity to parameters governing different parts of the process and its robustness to several model-error conditions. We find that the algorithm can achieve an acceptable level of error in many cases, with the weakest performance occurring for model-error cases and more extreme parameter regimes with more complex dynamics. Analysis of the poorest-performing cases indicates an initial decrease in error followed by an increase when the ensemble spread is reduced. Our results suggest avenues for further improvement through increasing ensemble spread by incorporating additive inflation or using a parameter or multi-model ensemble. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.
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Affiliation(s)
- Matthew J. Hoffman
- School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA
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Barone A, Fenton F, Veneziani A. Numerical sensitivity analysis of a variational data assimilation procedure for cardiac conductivities. CHAOS (WOODBURY, N.Y.) 2017; 27:093930. [PMID: 28964111 DOI: 10.1063/1.5001454] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
An accurate estimation of cardiac conductivities is critical in computational electro-cardiology, yet experimental results in the literature significantly disagree on the values and ratios between longitudinal and tangential coefficients. These are known to have a strong impact on the propagation of potential particularly during defibrillation shocks. Data assimilation is a procedure for merging experimental data and numerical simulations in a rigorous way. In particular, variational data assimilation relies on the least-square minimization of the misfit between simulations and experiments, constrained by the underlying mathematical model, which in this study is represented by the classical Bidomain system, or its common simplification given by the Monodomain problem. Operating on the conductivity tensors as control variables of the minimization, we obtain a parameter estimation procedure. As the theory of this approach currently provides only an existence proof and it is not informative for practical experiments, we present here an extensive numerical simulation campaign to assess practical critical issues such as the size and the location of the measurement sites needed for in silico test cases of potential experimental and realistic settings. This will be finalized with a real validation of the variational data assimilation procedure. Results indicate the presence of lower and upper bounds for the number of sites which guarantee an accurate and minimally redundant parameter estimation, the location of sites being generally non critical for properly designed experiments. An effective combination of parameter estimation based on the Monodomain and Bidomain models is tested for the sake of computational efficiency. Parameter estimation based on the Monodomain equation potentially leads to the accurate computation of the transmembrane potential in real settings.
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Affiliation(s)
- Alessandro Barone
- Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, USA
| | - Flavio Fenton
- Department of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
| | - Alessandro Veneziani
- Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia, USA; School of Advanced Studies IUSS, Pavia, Italy
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7
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Determining six cardiac conductivities from realistically large datasets. Math Biosci 2015; 266:15-22. [DOI: 10.1016/j.mbs.2015.05.008] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2014] [Revised: 05/20/2015] [Accepted: 05/22/2015] [Indexed: 11/17/2022]
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Alimohammadi M, Agu O, Balabani S, Díaz-Zuccarini V. Development of a patient-specific simulation tool to analyse aortic dissections: assessment of mixed patient-specific flow and pressure boundary conditions. Med Eng Phys 2013; 36:275-84. [PMID: 24290844 DOI: 10.1016/j.medengphy.2013.11.003] [Citation(s) in RCA: 48] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2013] [Revised: 10/10/2013] [Accepted: 11/06/2013] [Indexed: 10/26/2022]
Abstract
Aortic dissection has high morbidity and mortality rates and guidelines regarding surgical intervention are not clearly defined. The treatment of aortic dissection varies with each patient and detailed knowledge of haemodynamic and mechanical forces would be advantageous in the process of choosing a course of treatment. In this study, a patient-specific dissected aorta geometry is constructed from computed tomography scans. Dynamic boundary conditions are implemented by coupling a three element Windkessel model to the 3D domain at each outlet, in order to capture the essential behaviour of the downstream vasculature. The Windkessel model parameters are defined based on clinical data. The predicted minimum and maximum pressures are close to those measured invasively. Malperfusion is indicated and complex flow patterns are observed. Pressure, flow and wall shear stress distributions are analysed. The methodology presented here provides insight into the haemodynamics in a patient-specific dissected aorta and represents a development towards the use of CFD simulations as a diagnostic tool for aortic dissection.
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Affiliation(s)
- Mona Alimohammadi
- Department of Mechanical Engineering, University College London, WC1E 7JE, UK
| | | | - Stavroula Balabani
- Department of Mechanical Engineering, University College London, WC1E 7JE, UK
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A multi-electrode array and inversion technique for retrieving six conductivities from heart potential measurements. Med Biol Eng Comput 2013; 51:1295-303. [DOI: 10.1007/s11517-013-1101-2] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2013] [Accepted: 07/10/2013] [Indexed: 10/26/2022]
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Veneziani A, Vergara C. Inverse problems in Cardiovascular Mathematics: toward patient-specific data assimilation and optimization. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2013; 29:723-725. [PMID: 23825098 DOI: 10.1002/cnm.2566] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
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Estimating Electrical Conductivity Tensors of Biological Tissues Using Microelectrode Arrays. Ann Biomed Eng 2012; 40:2140-55. [DOI: 10.1007/s10439-012-0581-9] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2011] [Accepted: 04/21/2012] [Indexed: 10/28/2022]
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