Mathematical Cardiac Electrophysiology

Franzone, Piero Colli; Pavarino, Luca F.; Scacchi, Simone

doi:10.1007/978-3-319-04801-7

Cited by 189 publications

(278 citation statements)

References 410 publications

(727 reference statements)

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“…Tables 7, 8 report the CPU times, number of iterations, and system size for the FDM as the number of cells included in the simulation is increased. In Table 7, we use the passive model (10) for I ion , and in Table 8, we use the Grandi AP model. The linear systems are solved using the BiCGStab method (see [68,69]) with an incomplete LU preconditioner (e.g., [68]) and relative tolerance of 10 −5 for the true/unpreconditioned (l 2 ) norm of the residuum.…”

Section: Cpu Requirementsmentioning

confidence: 99%

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A Cell-Based Framework for Numerical Modeling of Electrical Conduction in Cardiac Tissue

et al. 2017

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In this paper, we study a mathematical model of cardiac tissue based on explicit representation of individual cells. In this EMI model, the extracellular (E) space, the cell membrane (M), and the intracellular (I) space are represented as separate geometrical domains. This representation introduces modeling flexibility needed for detailed representation of the properties of cardiac cells including their membrane. In particular, we will show that the model allows ion channels to be non-uniformly distributed along the membrane of the cell. Such features are difficult to include in classical homogenized models like the monodomain and bidomain models frequently used in computational analyses of cardiac electrophysiology. The EMI model is solved using a finite difference method (FDM) and two variants of the finite element method (FEM). We compare the three schemes numerically, reporting on CPU-efforts and convergence rates. Finally, we illustrate the distinctive capabilities of the EMI model compared to classical models by simulating monolayers of cardiac cells with heterogeneous distributions of ionic channels along the cell membrane. Because of the detailed representation of every cell, the computational problems that result from using the EMI model are much larger than for the classical homogenized models, and thus represent a computational challenge. However, our numerical simulations indicate that the FDM scheme is optimal in the sense that the computational complexity increases proportionally to the number of cardiac cells in the model. Moreover, we present simulations, based on systems of equations involving ∼117 million unknowns, representing up to ∼16,000 cells. We conclude that collections of cardiac cells can be simulated using the EMI model, and that the EMI model enable greater modeling flexibility than the classical monodomain and bidomain models.

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Section: Cpu Requirementsmentioning

confidence: 99%

“…These latter models are based on homogenization of the cardiac tissue and the extracellular space, the intracellular space, and the cell membrane are all assumed to exist everywhere (e.g., [9][10][11][12][13]). In the following, when we refer to homogenized models, we will refer to models of the monodomain or bidomain type.…”

Section: Introductionmentioning

confidence: 99%

A Cell-Based Framework for Numerical Modeling of Electrical Conduction in Cardiac Tissue

et al. 2017

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“…Therefore, even if we considered the total derivative of a generic concentration c (18) Dc Dt = ∂c ∂t + grad c · V, the second term would be negligible as the diffusive flow has already been removed. Now, it is well known that the Bidomain model in a one-dimensional fiber is equivalent to the Monodomain one (see [27,28]), which, for constant intra-and extracellular conductivity coefficients σ i,e f along the fiber, is given by (19) c m ∂v…”

Section: The Electrophysiological Modelmentioning

confidence: 99%

“…The extracellular conductivity σ e f has a conservative value of 2 mS/cm (see, e.g., [28]), whilst the intracellular one σ i f is computed as [29,30] (21)…”

Section: The Electrophysiological Modelmentioning

confidence: 99%

“…0.01 cm). We use the values r cyt = 150 Ω · cm and r junct = 1.5 Ω · cm 2 from [31] to obtain a value of 3 mS/cm for σ i f [28] if θ f = 1, which, together with the conservative one chosen for c m (1 µF/cm), ensures a conduction velocity between 0.06 and 0.07 cm/ms in case of the healthy fiber. Moreover, the same variable L cell affects the geometric plasma membrane area A geo = 2πR 2 cell + 2πR cell L cell (where R cell is the cardiomyocytes radius, i.e.…”

Section: The Electrophysiological Modelmentioning

confidence: 99%

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Computational modeling of the electromechanical response of a ventricular fiber affected by eccentric hypertrophy

Bianco

Franzone

Scacchi

et al. 2017

Communications in Applied and Industrial Mathematics

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The aim of this work is to study the effects of eccentric hypertrophy on the electromechanics of a single myocardial ventricular fiber by means of a one-dimensional finite-element strongly-coupled model. The electrical current flow model is written in the reference configuration and it is characterized by two geometric feedbacks, i.e. the conduction and convection ones, and by the mechanoelectric feedback due to stretchactivated channels. First, the influence of such feedbacks is investigated for both a healthy and a hypertrophic fiber in case of isometric simulations. No relevant discrepancies are found when disregarding one or more feedbacks for both fibers. Then, all feedbacks are taken into account while studying the electromechanical responses of fibers. The results from isometric tests do not point out any notable difference between the healthy and hypertrophic fibers as regards the action potential duration and conduction velocity. The length-tension relationships show increased stretches and reduced peak values for tension instead. The tension-velocity relationships derived from afterloaded isotonic and quickrelease tests depict higher values of contraction velocity at smaller afterloads. Moreover, higher maximum shortenings are achieved during the isotonic contraction. In conclusion, our simulation results are innovative in predicting the electromechanical behavior of eccentric hypertrophic fibers.

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Deep Learning‐Based Reduced Order Models for Cardiac Electrophysiology

Fresca,

Dedè,

Manzoni

2024

Big Data Analysis and Artificial Intelligence for Medical Sciences

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Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depolarization cycle occurring at every heart beat that models the time evolution of the electrical potential across the cell membrane, as well as a set of ionic variables. Multiple solutions of these systems, corresponding to different model inputs, are required to evaluate outputs of clinical interest, such as activation maps and action potential duration. More importantly, these models feature coherent structures that propagate over time, such as wavefronts. These systems can hardly be reduced to lower dimensional problems by conventional reduced order models (ROMs) such as, e.g., the reduced basis method. This is primarily due to the low regularity of the solution manifold (with respect to the problem parameters), as well as to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To overcome this difficulty, in this paper we propose a new, nonlinear approach relying on deep learning (DL) algorithms-such as deep feedforward neural networks and convolutional autoencodersto obtain accurate and efficient ROMs, whose dimensionality matches the number of system parameters. We show that the proposed DL-ROM framework can efficiently provide solutions to parametrized electrophysiology problems, thus enabling multi-scenario analysis in pathological cases. We investigate four challenging test cases in cardiac electrophysiology, thus demonstrating that DL-ROM outperforms classical projection-based ROMs.

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Mathematical Cardiac Electrophysiology

Cited by 189 publications

References 410 publications

A Cell-Based Framework for Numerical Modeling of Electrical Conduction in Cardiac Tissue

A Cell-Based Framework for Numerical Modeling of Electrical Conduction in Cardiac Tissue

Computational modeling of the electromechanical response of a ventricular fiber affected by eccentric hypertrophy

Deep Learning‐Based Reduced Order Models for Cardiac Electrophysiology

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