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We propose a boundary element method for the accurate solution of the cell-by-cell bidomain model of electrophysiology. The cell-by-cell model, also called Extracellular-Membrane-Intracellular (EMI) model, is a system of reaction–diffusion equations describing the evolution of the electric potential within each domain: intra- and extra-cellular space and the cellular membrane. The system is parabolic but degenerate because the time derivative is only in the membrane domain. In this work, we adopt a boundary-integral formulation for removing the degeneracy in the system and recast it to a parabolic equation on the membrane. The formulation is also numerically advantageous since the number of degrees of freedom is sensibly reduced compared to the original model. Specifically, we prove that the boundary-element discretization of the EMI model is equivalent to a system of ordinary differential equations, and we consider a time discretization based on the multirate explicit stabilized Runge–Kutta method. We numerically show that our scheme convergences exponentially in space for the single-cell case. We finally provide several numerical experiments of biological interest. [ABSTRACT FROM AUTHOR]

Computational models in cardiac electrophysiology are notorious for long runtimes, restricting the numbers of nodes and mesh elements in the numerical discretisations used for their solution. This makes it particularly challenging to incorporate structural heterogeneities on small spatial scales, preventing a full understanding of the critical arrhythmogenic effects of conditions such as cardiac fibrosis. In this work, we explore the technique of homogenisation by volume averaging for the inclusion of non-conductive micro-structures into larger-scale cardiac meshes with minor computational overhead. Importantly, our approach is not restricted to periodic patterns, enabling homogenised models to represent, for example, the intricate patterns of collagen deposition present in different types of fibrosis. We first highlight the importance of appropriate boundary condition choice for the closure problems that define the parameters of homogenised models. Then, we demonstrate the technique's ability to correctly upscale the effects of fibrotic patterns with a spatial resolution of 10 µm into much larger numerical mesh sizes of 100- 250 µm. The homogenised models using these coarser meshes correctly predict critical pro-arrhythmic effects of fibrosis, including slowed conduction, source/sink mismatch, and stabilisation of re-entrant activation patterns. As such, this approach to homogenisation represents a significant step towards whole organ simulations that unravel the effects of microscopic cardiac tissue heterogeneities. • Upscaling for feasible simulation of microscale fibrosis in cardiac electrophysiology • Practical consideration of upscaling for dynamics of sharp-fronted travelling waves • Demonstrated recovery of pro-arrhythmic effects of fibrosis in upscaled models • Selection of boundary conditions to balance numerical and upscaling errors • Significant speed-up for monodomain model simulation in the presence of microfibrosis [ABSTRACT FROM AUTHOR]

Electrotonic interactions between cardiac cells modulate the dispersion of action potential duration (APD). This paper provides a complete mathematical analysis of a simple model of exponential-shaped repolarization in a network of electrotonically-coupled cells with different intrinsic APDs. The forward problem consists in computing the APD map in the coupled system from the intrinsic APD map. A closed-form algebraic formula is derived for the forward problem. The inverse problem, inferring the intrinsic APDs from an APD map, is proved to have a unique solution (if any). Perturbation analysis leads to an efficient and accurate Newton-based solver for this specific inverse problem. Finally, an analytical expression is obtained for the convolution filter that solves the forward problem in one dimension. This mathematical framework forms a solid theoretical basis for future development and validation of repolarization parameter estimation techniques in detailed models of cardiac tissue. [ABSTRACT FROM AUTHOR]

Abstract: This work considers the approximation of the cardiac bidomain equations, either isolated or coupled with the torso, via first order semi-implicit time-marching schemes involving a fully decoupled computation of the unknown fields (ionic state, transmembrane potential, extracellular and torso potentials). For the isolated bidomain system, we show that the Gauss–Seidel and Jacobi like splittings do not compromise energy stability; they simply alter the energy norm. Within the framework of the numerical simulation of electrocardiograms (ECG), these bidomain splittings are combined with an explicit Robin–Robin treatment of the heart–torso coupling conditions. We show that the resulting schemes allow a fully decoupled (energy) stable computation of the heart and torso fields, under an additional hyperbolic-CFL like condition. The accuracy and convergence rate of the considered schemes are investigated numerically with a series of numerical experiments. [Copyright &y& Elsevier]

Abstract: Chaste (‘Cancer, heart and soft-tissue environment’) is a software library and a set of test suites for computational simulations in the domain of biology. Current functionality has arisen from modelling in the fields of cancer, cardiac physiology and soft-tissue mechanics. It is released under the LGPL 2.1 licence. Chaste has been developed using agile programming methods. The project began in 2005 when it was reasoned that the modelling of a variety of physiological phenomena required both a generic mathematical modelling framework, and a generic computational/simulation framework. The Chaste project evolved from the Integrative Biology (IB) e-Science Project, an inter-institutional project aimed at developing a suitable IT infrastructure to support physiome-level computational modelling, with a primary focus on cardiac and cancer modelling. Program summary: Program title: Chaste Catalogue identifier: AEFD_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEFD_v1_0.html Program obtainable from: CPC Program Library, Queen''s University, Belfast, N. Ireland Licensing provisions: LGPL 2.1 No. of lines in distributed program, including test data, etc.: 5 407 321 No. of bytes in distributed program, including test data, etc.: 42 004 554 Distribution format: tar.gz Programming language: C++ Operating system: Unix Has the code been vectorised or parallelized?: Yes. Parallelized using MPI. RAM: for two of the scenarios described in Section 6 of the manuscript (Monodomain re-entry on a slab or Cylindrical crypt simulation). Up to 16 Gigabytes (distributed across processors) for full resolution bidomain cardiac simulation. Classification: 3. External routines: Boost, CodeSynthesis XSD, CxxTest, HDF5, METIS, MPI, PETSc, Triangle, Xerces Nature of problem: Chaste may be used for solving coupled ODE and PDE systems arising from modelling biological systems. Use of Chaste in two application areas are described in this paper: cardiac electrophysiology and intestinal crypt dynamics. Solution method: Coupled multi-physics with PDE, ODE and discrete mechanics simulation. Running time: The largest cardiac simulation described in the manuscript takes about 6 hours to run on a single 3 GHz core. See results section (Section 6) of the manuscript for discussion on parallel scaling. [Copyright &y& Elsevier]

Abstract: The simulation of cardiac electrophysiological waves are known to require extremely fine meshes, limiting the applicability of current numerical models to simplified geometries and ionic models. In this work, an accurate numerical method based on a time-dependent anisotropic remeshing strategy is presented for simulating three-dimensional cardiac electrophysiological waves. The proposed numerical method greatly reduces the number of elements and enhances the accuracy of the prediction of the electrical wave fronts. Illustrations of the performance and the accuracy of the proposed method are presented using a realistic heart geometry. Qualitative and quantitative results show that the proposed methodology is far superior to the uniform mesh methods commonly used in cardiac electrophysiology. [Copyright &y& Elsevier]

Abstract: The steady states of the Fenton–Karma, the Courtemanche and the Nygren cell models were studied by determining the fixed points of the dynamical system describing their cell kinetics. The linear stability of the fixed points was investigated, as well as their response to external stimuli. Symbolic calculations were carried out as far as possible in order to prove the existence of these fixed points. In the Fenton–Karma model, a unique stable fixed point was found, namely the resting state. In contrast, the Courtemanche model had an infinite number of fixed points. A bifurcation diagram was constructed by classifying these fixed points according to a conservation law. Initial conditions were identified, for which the dynamical behavior of the cell was auto-oscillatory. In its original formulation, the Nygren model had no fixed point. After having restored charge conservation, the system was found to have an infinite number of fixed points, resulting in a bifurcation diagram similar to that of the Courtemanche model. The approach proposed in this paper assists in the exploration of the high-dimensional parameter space of the cell models and the identification of the conditions leading to spontaneous pacemaker activity. [Copyright &y& Elsevier]

• Fractional-order differential equations models can simulate non-ideal electrical circuit components. • Cardiac electrical models with non-ideal circuit components account for physiological "capacitive memory" effects. • Electrical instabilities known as alternans can arise in cardiac cells via voltage- or calcium-mediated mechanisms. • Capacitive memory can either suppress or promote alternans depending on the underlying mechanism. Cardiac myocyte electrical activity is traditionally approximated with ideal resistor-capacitor circuit networks. However, non-ideal circuit components may provide a more realistic approximation of excitable cell behavior. Such non-ideal circuit components are governed by fractional-order dynamics and contribute capacitive memory effects to the excitable cell system. Our prior work has detailed the effects of cell membrane-derived capacitive memory in a minimal cardiac model driven by voltage instabilities, and capacitive memory has been shown to shorten the action potential duration (APD) and suppress a beat-to-beat alternation in the APD known as alternans. In this study, we investigate the effects of memory in a biophysically detailed cardiac model that accounts for detailed representations of intracellular calcium cycling and transmembrane voltage dynamics. We perform simulations of varying fractional-order and pacing cycle length and investigate conditions in which alternans is driven by either voltage- or calcium-mediated instabilities. We found that capacitive memory suppresses alternans when calcium-mediated. Interestingly, when mediated by voltage-driven instabilities, memory effects induced a calcium instability that in turn promoted alternans under most conditions. In summary, capacitive memory due to fractional-order dynamics alters electrical signaling in cardiac cells in a manner than may either promote or suppress instabilities. [ABSTRACT FROM AUTHOR]