Monodomain Simulation

In this paper, we find relations between the ionic parameters and the diffusion parameters which are sufficient to ensure the existence of a periodic solution for a well-known monodomain model in a weak sense. We make use of the method of approximation of Faedo-Galerkin to prove the existence of weak periodic solutions of the monodomain model for the electrical activity of the heart assuming that it is periodically activated in its boundaries. Actually, this periodic solution has the same period of activation. Finally, we reflect on how these ionic-diffusive relations are useful to explain the pathophysiology of some rhythm disorders.

In this paper, we find relations between the ionic parameters and the diffusion parameters which are sufficient to ensure the existence of a periodic solution for a well-known monodomain model in a weak sense. We make use of the method of approximation of Faedo-Galerkin to prove the existence of weak periodic solutions of the monodomain model for the electrical activity of the heart assuming that it is periodically activated in its boundaries. Actually, this periodic solution has the same period of activation. Finally, we reflect on how these ionic-diffusive relations are useful to explain the pathophysiology of some rhythm disorders.

In this paper, we investigate the convergence of the Faedo-Galerkin approximations, in a strong sense, to a strong T-periodic solution of the torso-coupled bidomain model where T is the period of activation of the inner wall of heart. First, we define the torso-coupled bi-domain operator and prove some of its more important properties for our work. After, we define the abstract evolution system of equations associated with torso-coupled bidomain model and give the definition of strong solution. We prove that the Faedo-Galerkin's approximations have the regularity of a strong solution, and we find that some restrictions can be imposed over the initial conditions, so that this sequence of Faedo-Galerkin fully converge to a strong solution of the Cauchy problem. Finally, this results are used for showing the existence a strong T-periodic solution.

Se formula un modelo de monodominio de actividad eléctrica en un ventrículo aislado. Este modelo se escribe como una EDP de tipo reacción difusión acoplada a una EDO, se utiliza el modelo de Rogers-Mculloch para representar la actividad eléctrica a través de la membrana celular. Método: Se proponen definiciones de solución débil y fuerte respectivamente para el problema de Cauchy variacional asociado al modelo de monodominio. Se propone una sucesión de soluciones aproximadas de tipo Faedo-Galerkin. Resultados: Se demuestra que la sucesión de soluciones aproximadas converge a una solución débil según la definición que se propone. Finalmente, se obtiene que la solución débil es también una solución fuerte. Conclusión: El modelo de monodominio de actividad eléctrica en un ventrículo aislado que se propone tiene solución débil en un sentido apropiado. Además, esta solución débil también es una solución fuerte.

In this paper, we investigate the convergence of the Faedo-Galerkin approximations, in a strong sense, to a strong T-periodic solution of the torso-coupled bidomain model where T is the period of activation of the inner wall of the heart. First, we define the torso-coupled bidomain operator and prove some of its more important properties for our work. After, we define the abstract evolution system of the equations that are associated with torso-coupled bidomain model and give the definition of a strong solution. We prove that the Faedo-Galerkin’s approximations have the regularity of a strong solution, and we find that some restrictions can be imposed over the initial conditions, so that this sequence of Faedo-Galerkin fully converges to a strong solution of the Cauchy problem. Finally, these results are used for showing the existence a strong T-periodic solution.

In this paper, we obtain a result of existence of weak periodic solution of the model of monodomain for the electrical activity of an isolated ventricle considering that it is activated periodically. In this regard, this periodic solution has the same period of the activation. We use the Faedo-Galerkin scheme to approximate the periodic solution and we prove that each Faedo-Galerkin system of ordinary differential equations has a periodic solution, via a fixed point theorem. Later, it is proved that this sequence of approximations converges, in a suitable sense, to a weak period solution of the monodomain model.

Ongoing developments in cardiac modelling have resulted, in particular, in the development of advanced and increasingly complex computational frameworks for simulating cardiac tissue electrophysiology. The goal of these simulations is often to represent the detailed physiology and pathologies of the heart using codes that exploit the computational potential of high-performance computing architectures. These developments have rapidly progressed the simulation capacity of cardiac virtual physiological human style models; however, they have also made it increasingly challenging to verify that a given code provides a faithful representation of the purported governing equations and corresponding solution techniques. This study provides the first cardiac tissue electrophysiology simulation benchmark to allow these codes to be verified. The benchmark was successfully evaluated on 11 simulation platforms to generate a consensus gold-standard converged solution. The benchmark definition in c...

We discuss the application of the spectral element method to the monodomain and bidomain equations describing propagation of cardiac action potential. Models of cardiac electrophysiology consist of a system of partial differential equations coupled with a system of ordinary differential equations representing cell membrane dynamics. The solution of these equations requires solving multiple length scales due to the ratio of advection to diffusion that varies among the different equations. High order approximation of spectral elements provides greater flexibility in resolving multiple length scales. Furthermore, spectral elements are extremely efficient to model propagation phenomena on complex shapes using fewer degrees of freedom than its finite element equivalent (for the same level of accuracy). We illustrate a fully unstructured all-hexahedra approach implementation of the method and we apply it to the solution of full 3D monodomain and bidomain test cases. We discuss some key el...

Ongoing developments in cardiac modelling have resulted, in particular, in the development of advanced and increasingly complex computational frameworks for simulating cardiac tissue electrophysiology. The goal of these simulations is often to represent the detailed physiology and pathologies of the heart using codes that exploit the computational potential of high-performance computing architectures. These developments have rapidly progressed the simulation capacity of cardiac virtual physiological human style models; however, they have also made it increasingly challenging to verify that a given code provides a faithful representation of the purported governing equations and corresponding solution techniques. This study provides the first cardiac tissue electrophysiology simulation benchmark to allow these codes to be verified. The benchmark was successfully evaluated on 11 simulation platforms to generate a consensus gold-standard converged solution. The benchmark definition in c...

In this paper, we obtain a result of existence of weak periodic solution of the model of monodomain for the electrical activity of an isolated ventricle considering that it is activated periodically. In this regard, this periodic solution has the same period of the activation. We use the Faedo-Galerkin scheme to approximate the periodic solution and we prove that each Faedo-Galerkin system of ordinary differential equations has a periodic solution, via a fixed point theorem. Later, it is proved that this sequence of approximations converges, in a suitable sense, to a weak period solution of the monodomain model.

This work concerns the unilateral problem for the Klein-Gordon operator L = ∂ 2 u ∂t 2 − M (|∇u| 2)∆u + M 1 (|u| 2)u − f. Using an appropriate penalization, we obtain a variational inequality for a perturbed equation, and then show the existence and uniqueness of solutions.

In this paper we consider linear operators of the form L + λI between suitable functions spaces, when 0 is an eigenvalue of L with constant eigenfunctions. We introduce a new notion of "quasi" uniform maximum principle, named k-uniform maximum principle, which holds for λ belonging to certain neighborhoods of 0 depending on k ∈ R +. Our approach is based on a L ∞ − L 2 estimate, which let us prove some generalization of known results for elliptic and parabolic problems with Neumann or periodic boundary conditions.

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue. They are especially important for accurately modelling extracellular stimulation, as evidenced by their prediction of virtual electrode polarization before experimental verification. However, solution of the equations is computationally expensive due to the fine spatial and temporal discretization needed. This limits the size and duration of the problem which can be modeled. Regardless of the specific form into which they are cast, the computational bottleneck becomes the repeated solution of a large, linear system. The purpose of this review is to give an overview of the equations, and the methods by which they have been solved. Of particular note are recent developments in multigrid methods, which have proven to be the most efficient.

A recent verification study compared 11 large-scale cardiac electrophysiology solvers on an unambiguously defined common problem. An unexpected amount of variation was observed between the codes, including significant error in conduction velocity in the majority of the codes at certain spatial resolutions. In particular, the results of the six finite element codes varied considerably despite each using the same order of interpolation. In this present study, we compare various algorithms for cardiac electrophysiological simulation, which allows us to fully explain the differences between the solvers. We identify the use of mass lumping as the fundamental cause of the largest variations, specifically the combination of the commonly used techniques of mass lumping and operator splitting, which results in a slightly different form of mass lumping to that supported by theory and leads to increased numerical error. Other variations are explained through the manner in which the ionic current is interpolated. We also investigate the effect of different forms of mass lumping in various types of simulation.

Compressibility influences the mechanical bidomain model describing the elastic properties of tissue. Displacements of the intracellular and extracellular spaces are analyzed individually, and differences in these displacements produce membrane forces. Two length constants are associated with the membrane spring constant, one contains the shear moduli and the other contains the bulk moduli. The analytical solutions in these examples indicate that the monodomain part does not contribute to the membrane force. Accounting for compressibility has its largest impact on the intracellular and extracellular pressures. The bidomain contribution to the pressure obeys the Helmholtz equation rather than Laplace's equation. This model predicts membrane forces that might cause tissue remodeling or mechanotransduction.

We introduce a preconditioner for the solution of the Bidomain system governing the propagation of action potentials in the myocardial tissue. The Bidomain model is a degenerate parabolic set of nonlinear reaction-diffusion equations. The nonlinear term describes the ion flux at the cellular level. The degenerate nature of the problem results in a severe ill conditioning of its discretization. Our preconditioning strategy is based on a suitable adaptation of the Monodomain model, a simplified version of the Bidomain one, which is by far simpler to solve, nevertheless is unable to capture significant features of the action potential propagation. The Monodomain preconditioner application to a non-symmetric formulation of the Bidomain system results at the algebraic level in a lower block-triangular preconditioner. We prove optimality of the preconditioner with respect to the mesh size, and corroborate our theoretical results with 3D numerical simulations both on idealized and real ventricle geometries.

The simulation of cardiac electrophysiology is a mature field in computational physiology. Recent advances in medical imaging, high-performance computing and numerical methods mean that computational models of electrical propagation in human heart tissue are ripe for use in patient-specific simulation for diagnosis, for prognosis and for selection of treatment methods. However, in order to move in this direction, it is necessary to make efficient use of modern petascale computing resources. This paper focuses on an existing open source simulation framework (Chaste) and documents work done to improve the parallel scaling on a small range of electrophysiology benchmark problems. These benchmarks involve the numerical solution of the monodomain or bidomain equations via the finite-element method. At the beginning of this study the electrophysiology libraries within Chaste were already enabled to run in parallel and were able to solve for electrical propagation using the monodomain or b...

We discuss the application of the spectral element method to the monodomain and bidomain equations describing propagation of cardiac action potential. Models of cardiac electrophysiology consist of a system of partial differential equations coupled with a system of ordinary differential equations representing cell membrane dynamics. The solution of these equations requires solving multiple length scales due to the ratio of advection to diffusion that varies among the different equations. High order approximation of spectral elements provides greater flexibility in resolving multiple length scales. Furthermore, spectral elements are extremely efficient to model propagation phenomena on complex shapes using fewer degrees of freedom than its finite element equivalent (for the same level of accuracy). We illustrate a fully unstructured all-hexahedra approach implementation of the method and we apply it to the solution of full 3D monodomain and bidomain test cases. We discuss some key el...

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 ...

In this paper, we obtain a result of existence of weak periodic solution of the model of monodomain for the electrical activity of an isolated ventricle considering that it is activated periodically. In this regard, this periodic solution has the same period of the activation. We use the Faedo-Galerkin scheme to approximate the periodic solution and we prove that each Faedo-Galerkin system of ordinary differential equations has a periodic solution, via a fixed point theorem. Later, it is proved that this sequence of approximations converges, in a suitable sense, to a weak period solution of the monodomain model.

Bidomain or monodomain modelling has been used widely to study various issues related to action potential propagation in cardiac tissue. In most of these previous studies, the finite difference method is used to solve the partial differential equations associated with the model. Though the finite difference approach has provided useful insight in many cases, adequate discretisation of cardiac tissue with realistic dimensions often requires a large number of nodes, making the numerical solution process difficult or impossible with available computer resources. Here, a Chebyshev pseudospectral method is presented that allows a significant reduction in the number of nodes required for a given solution accuracy. The new method is used to solve the governing nonlinear partial differential equation for the monodomain model representing a two-dimensional homogeneous sheet of cardiac tissue. The unknown transmembrane potential is expanded in terms of Chebyshev polynomial trial functions and the equation is enforced at the Gauss-Lobatto grid points. Spatial derivatives are obtained using the fast Fourier transform and the solution is advanced in time using an explicit technique. Numerical results indicate that the pseudospectral approach allows the number of nodes to be reduced by a factor of sixteen, while still maintaining the same error performance. This makes it possible to perform simulations with the same accuracy using about twelve times less CPU time and memory.

 Description of a method coupling parallelization in time and space on the GPU.  Resolution of a model describing the propagation of the electrical signal in a neural tissue using the parallel algorithm on GPUs.  Significant reduction in the calculation time to obtain an accurate solution.

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue. They are especially important for accurately modelling extracellular stimulation, as evidenced by their prediction of virtual electrode polarization before experimental verification. However, solution of the equations is computationally expensive due to the fine spatial and temporal discretization needed. This limits the size and duration of the problem which can be modeled. Regardless of the specific form into which they are cast, the computational bottleneck becomes the repeated solution of a large, linear system. The purpose of this review is to give an overview of the equations, and the methods by which they have been solved. Of particular note are recent developments in multigrid methods, which have proven to be the most efficient.

Computer simulations of electrical behaviour in the whole ventricles have become commonplace during the last few years. The goals of this article are (i) to review the techniques that are currently employed to model cardiac electrical activity in the heart, discussing the strengths and weaknesses of the various approaches, and (ii) to implement a novel modelling approach, based on physiological reasoning, that lifts some of the restrictions imposed by current state-of-the-art ionic models. To illustrate the latter approach, the present study uses a recently developed ionic model of the ventricular myocyte that incorporates an excitation–contraction coupling and mitochondrial energetics model. A paradigm to bridge the vastly disparate spatial and temporal scales, from subcellular processes to the entire organ, and from sub-microseconds to minutes, is presented. Achieving sufficient computational efficiency is the key to success in the quest to develop multiscale realistic models that...

Computer models offer many attractive benefits. However, the modeling of cardiac tissue is computationally expensive due to several physical constraints which result in fine spatiotemporal discretization over large spatiotemporal regions. Our laboratory has been actively trying to develop new techniques to make large scale cardiac simulations tractable over the past 15 years. This paper describes the latest modeling software that our group has developed, called Carp (Cardiac arrhythmias research package). It is designed to run in both shared memory and clustered computing environments. Carp aims to be modular and flexible by following a plug-in framework. This allows the latest models and most efficient solvers to be incorporated as well as enabling run-time selection of techniques. Performance results are given for a large-scale simulation which utilized a comprehensive membrane ionic current description.

In this paper, we obtain a result of existence of weak periodic solution of the model of monodomain for the electrical activity of an isolated ventricle considering that it is activated periodically. In this regard, this periodic solution has the same period of the activation. We use the Faedo-Galerkin scheme to approximate the periodic solution and we prove that each Faedo-Galerkin system of ordinary differential equations has a periodic solution, via a fixed point theorem. Later, it is proved that this sequence of approximations converges, in a suitable sense, to a weak period solution of the monodomain model.

Introducción: Se formula un modelo de monodominio de actividad eléctrica en un ventrículo aislado. Este modelo se escribe como una EDP de tipo reacción difusión acoplada a una EDO, se utiliza el modelo de Rogers-Mculloch para representar la actividad eléctrica a través de la membrana celular. Método: Se proponen definiciones de solución débil y fuerte respectivamente para el problema de Cauchy variacional asociado al modelo de monodominio. Se propone una sucesión de soluciones aproximadas de tipo Faedo-Galerkin. Resultados: Se demuestra que la sucesión de soluciones aproximadas converge a una solución débil según la definición que se propone. Finalmente, se obtiene que la solución débil es también una solución fuerte. Conclusión: El modelo de monodominio de actividad eléctrica en un ventrículo aislado que se propone tiene solución débil en un sentido apropiado. Además, esta solución débil también es una solución fuerte.

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue. They are especially important for accurately modeling extracellular stimulation, as evidenced by their prediction of virtual electrode polarization before experimental verification. However, solution of the equations is computationally expensive due to the fine spatial and temporal discretization needed. This limits the size and duration of the problem which can be modeled. Regardless of the specific form into which they are cast, the computational bottleneck becomes the repeated solution of a large, linear system. The purpose of this review is to give an overview of the equations and the methods by which they have been solved. Of particular note are recent developments in multigrid methods, which have proven to be the most efficient.

Computer models offer many attractive benefits. However, the modeling of cardiac tissue is computationally expensive due to several physical constraints which result in fine spatiotemporal discretization over large spatiotemporal regions. Our laboratory has been actively trying to develop new techniques to make large scale cardiac simulations tractable over the past 15 years. This paper describes the latest modeling software that our group has developed, called Carp (Cardiac arrhythmias research package). It is designed to run in both shared memory and clustered computing environments. Carp aims to be modular and flexible by following a plug-in framework. This allows the latest models and most efficient solvers to be incorporated as well as enabling run-time selection of techniques. Performance results are given for a large-scale simulation which utilized a comprehensive membrane ionic current description.

Many problems in Biology and Engineering are governed by anisotropic reaction-diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately capture the propagating wave while avoiding the appearance of spurious oscillations in the wave front. This work develops a family of macro finite elements amenable for solving anisotropic reaction-diffusion equations with stiff reactive terms. The developed elements are incorporated on a semi-implicit algorithm based on operator splitting that includes adaptive time stepping for handling the stiff reactive term. A linear system is solved on each time step to update the transmembrane potential, whereas the remaining ordinary differential equations are solved uncoupled. The method allows solving the linear system on a coarser mesh thanks to the static condensation of the internal degrees of freedom (DOF) of the macroelements while maintaining the accuracy of the finer mesh. The method and algorithm have been implemented in parallel. The accuracy of the method has been tested on two-and three-dimensional examples demonstrating excellent behavior when compared to standard linear elements. The better performance and scalability of different macro finite elements against standard finite elements have been demonstrated in the simulation of a human heart and a heterogeneous two-dimensional problem with reentrant activity. Results have shown a reduction of up to four times in computational cost for the macro finite elements with respect to equivalent (same number of DOF) standard linear finite elements as well as good scalability properties.

Numerical simulations of defibrillation using the Bidomain model coupled to a model of membrane kinetics represent a serious numerical challenge. This is because very high voltages close to defibrillation electrodes demand that extreme time step restrictions be placed on standard numerical schemes, e.g. the forward Euler scheme. A common solution to this problem is to modify the cell model by simple if-tests applied to several equations and rate functions. These changes are motivated by numerical problems rather than physiology, and should therefore be avoided whenever possible.

The bidomain equations are widely accepted to model the spatial distribution of the electrical potential in the heart. Although order optimal methods have been devised for discrete versions of these equations, it is still very CPU demanding to solve the equations numerically on a sufficiently fine mesh in 3D. Furthermore, the equations are hard to analyze; from a mathematical point of view, very little is known about the qualitative behavior of the solutions generated by these equations.

In this study, a computational framework combining state-of-the-art cardiac simulation software and Grid computing is used to investigate the impact of the block of the HERG current on the ECG waveform using state-of-the-art 3D ventricular models of electrophysiology. The technology developed enables (i) automated parameter sweeping using multiscale models (from ion channel to ECG) and (ii) reduced execution time of the simulations performed.

Advanced medical imaging technologies provide a wealth of information on cardiac anatomy and structure at a paracellular resolution, allowing to identify microstructural discontinuities which disrupt the intracellular matrix. Current stateof-the-art computer models built upon such datasets account for increasingly finer anatomical details, however, structural discontinuities at the paracellular level are typically discarded in the model generation process, owing to the significant costs which incur when using high resolutions for explicit representation. In this study, a novel discontinuous finite element (dFE) approach for discretizing the bidomain equations is presented, which accounts for fine-scale structures in a computer model without the need to increase spatial resolution. In the dFE method, this is achieved by imposing infinitely thin lines of electrical insulation along edges of finite elements which approximate the geometry of discontinuities in the intracellular matrix. Simulation results demonstrate that the dFE approach accounts for effects induced by microscopic size scale discontinuities, such as the formation of microscopic virtual electrodes, with vast computational savings as compared to high resolution continuous finite element models. Moreover, the method can be implemented in any standard continuous finite element code with minor effort.

A bidomain reaction-diffusion model of the human heart was developed, and potentials resulting from normal depolarization and repolarization were compared with results from a compatible monodomain model. Comparisons were made for an empty isolated heart and for a heart with fluid-filled ventricles. Both sinus rhythm and ectopic activation were simulated. The bidomain model took 2 days on 32 processors to simulate a complete cardiac cycle. Differences between monodomain and bidomain results were extremely small, even for the extracellular potentials, which in case of the monodomain model were computed with a high-resolution forward model. Propagation of activation was 2% faster in the bidomain model than in the monodomain model. Electrograms computed with monodomain and bidomain models were visually indistinguishable. We conclude that, in the absence of applied currents, propagating action potentials on the scale of a human heart can be studied with a monodomain model.

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) that describe the ionic currents at the myocardial cell level. Generating an efficient numerical solution of these ODEs is a challenging task, and, in fact, the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this paper, we examine the efficiency of the numerical solution of four cardiac electrophysiological models using implicit--explicit Runge-Kutta (IMEX-RK) splitting methods. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform the methods commonly used in practice.

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.

A bidomain reaction-diffusion model of the human heart was developed, and potentials resulting from normal depolarization and repolarization were compared with results from a compatible monodomain model. Comparisons were made for an empty isolated heart and for a heart with fluid-filled ventricles. Both sinus rhythm and ectopic activation were simulated. The bidomain model took 2 days on 32 processors to simulate a complete cardiac cycle. Differences between monodomain and bidomain results were extremely small, even for the extracellular potentials, which in case of the monodomain model were computed with a high-resolution forward model. Propagation of activation was 2% faster in the bidomain model than in the monodomain model. Electrograms computed with monodomain and bidomain models were visually indistinguishable. We conclude that, in the absence of applied currents, propagating action potentials on the scale of a human heart can be studied with a monodomain model.

A generalized finite difference (GFD) method is presented that can be used to solve the bidomain equations modeling cardiac electrical activity. Classical finite difference methods have been applied by many researchers to the bidomain equations. However, these methods suffer from the limitation of requiring computational meshes that are structured and orthogonal. Finite element or finite volume methods enable the bidomain equations to be solved on unstructured meshes, although implementations of such methods do not always cater for meshes with varying element topology. The GFD method solves the bidomain equations on arbitrary and irregular computational meshes without any need to specify element basis functions. The method is useful as it can be easily applied to activation problems using existing meshes that have originally been created for use by finite element or finite difference methods. In addition, the GFD method employs an innovative approach to enforcing nodal and non-nodal boundary conditions. The GFD method performs effectively for a range of two and three-dimensional test problems and when computing bidomain electrical activation moving through a fully anisotropic three-dimensional model of canine ventricles.

This paper describes a finite volume method for modeling electrical activation in a sample of cardiac tissue using the bidomain equations. Microstructural features to the level of cleavage planes between sheets of myocardial fibres in the tissue are explicitly represented. The key features of this implementation compared to previous modeling are that it is designed to represent physical discontinuities with no volume in the intracellular domain and to provide linear systems of equations that are computationally efficient to construct and solve. Results obtained using this method highlight how the understanding of discontinuous activation in cardiac tissue can form a basis for better understanding defibrillation processes and experimental recordings.

Chaste -Cancer, Heart And Soft Tissue Environment -is an open source C++ library for the computational simulation of mathematical models developed for physiology and biology. Code development has been driven by two initial applications: cardiac electrophysiology and cancer development. A large number of cardiac electrophysiology studies have been enabled and performed, including high-performance computational investigations of defibrillation on realistic human cardiac geometries. New models for the initiation and growth of tumours have been developed. In particular, cellbased simulations have provided novel insight into the role of stem cells in the colorectal crypt. Chaste is constantly evolving and is now being applied to a far wider range of problems. The code provides modules for handling common scientific computing components, such as meshes and solvers for ordinary and partial differential equations (ODEs/PDEs). Re-use of these components avoids the need for researchers to 're-invent the wheel' with each new project, accelerating the rate of progress in new applications. Chaste is developed using industrially-derived techniques, in particular testdriven development, to ensure code quality, re-use and reliability. In this article we provide examples that illustrate the types of problems Chaste can be used to solve, which can be run on a desktop computer. We highlight some scientific studies that have used or are using Chaste, and the insights they have provided. The source code, both for specific releases and the development version, is available to download under an open source Berkeley Software Distribution (BSD) licence at

Cardiac modelling is the area of physiome modelling where the available simulation software is perhaps most mature, and it therefore provides an excellent starting point for considering the software requirements for the wider Physiome community. In this paper, we will begin by introducing some of the most advanced existing software packages for simulating cardiac electrical activity. We consider the software development methods used in producing codes of this type, and discuss their use of numerical algorithms, relative computational efficiency, usability, robustness and extensibility. We then go on to describe a class of software development methodologies known as test-driven agile methods and argue that such methods are more suitable for scientific software development than the traditional academic approaches. A case study is a project of our own, Chaste (Cancer, Heart And Soft Tissue Environment), which is a library of computational biology software that began as an experiment in the use of agile programming methods. We present our experiences with a review of our progress thus far, focusing on the advantages and disadvantages of this new approach compared to the development methods used in some existing packages.

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.

We present an application of high order hierarchical finite elements for the efficient approximation of solutions to the cardiac monodomain problem. We detail the hurdles which must be overcome in order to achieve theoretically-optimal errors in the approximations generated, including the choice of method for approximating the solution to the cardiac cell model component. We place our work on a solid theoretical foundation and show that it can greatly improve the accuracy in the approximation which can be achieved in a given amount of processor time. Our results demonstrate superior accuracy over linear finite elements at a cheaper computational cost and thus indicate the potential indispensability of our approach for large-scale cardiac simulation. solution techniques, including work on preconditioning, parallelisation and adaptivity in space and time . In this study, we investigate the potential of reducing the number of DOF by using a high-order polynomial FEM to approximate the monodomain PDE in space, with the goal of significantly improving simulation efficiency over the piecewise-linear FEM approach commonly used in the field . For schemes where the polynomial degree p of the elements is adjusted according to the error in the approximation, this is known as the finite element p-version. In the work presented here, we work with schemes which keep p fixed.

Although the cardiac bidomain model has been widely used in the simulation of electrical activation, its relatively computationally expensive nature means that monodomain approaches are generally required for long duration simulations (for example, investigations of arrhythmia mechanisms). However, the presence of a conducting bath surrounding the tissue is known to induce wavefront curvature (surface leading bulk), a phenomena absent in standard monodomain approaches. Here, we investigate the biophysical origin of the bidomain bath-loading induced wavefront curvature and present a novel augmented monodomain equivalent bidomain approach faithfully replicating all aspects of bidomain wavefront morphology and conduction velocity, but with a fraction of the computational cost. Bath-loading effects are shown to be highly dependent upon specific conductivity parameters, but less dependent upon the thickness or conductivity of the surrounding bath, with even relatively thin surrounding fluid layers (~ 0.1 mm) producing significant wavefront curvature in bidomain simulations. We demonstrate that our augmented monodomain approach can be easily adapted for different conductivity sets and applied to anatomically complex models thus facilitating fast and accurate simulation of cardiac wavefront dynamics during long duration simulations, further aiding the faithful comparison of simulations with experiments.

has described the software engineering and computational infrastructure that has been set up as part of the Chaste project. Chaste-Cancer Heart And Soft Tissue Environment-is an open source software package that currently has heart and cancer modelling functionality. This software has been written using a programming paradigm imported from the commercial sector, and has resulted in code that has been subject to a far more rigorous testing procedure than is usual in this field. The exceptionally large number of tests written for each small, individual subroutine of this code allows software development to be carried out with confidence that new developments do not inadvertently degrade earlier functionality. This software may be developed for two main purposes-either to add new physiological functionality, or to incorporate a different numerical algorithm for solving the governing equations which allows a given level of accuracy to be attained more efficiently. In this paper, we will focus on the second of these purposes for the field of cardiac electrophysiological modelling. Whiteley (Annals of Biomedical Engineering, vol. 36, pp. 1398-1408) has developed a numerical algorithm for solving the bidomain equations that utilizes the multi-scale nature of the physiology modelled to enhance computational efficiency. When using this algorithm, only quantities that vary on short time-scales or short length-scales are computed to a fine resolution. Other more slowly varying quantities are computed at a lower resolution. Interpolation is then used to transfer information between different resolutions. Using a simple geometry in two dimensions and a purpose-built code, this algorithm was reported to give an increase in computational efficiency of over two orders of magnitude. In this paper, we begin by reviewing numerical methods currently in use for solving the bidomain equations, explaining how these methods may be developed to utilize the multi-scale algorithm discussed above. We then demonstrate the use of this algorithm within the Chaste framework for solving the monodomain and bido- † Author for correspondence (jonathan.whiteley@comlab.ox.ac.uk) main equations in a three-dimensional realistic heart geometry. Finally, we discuss how Chaste may be developed to include new physiological functionality-such as modelling a beating heart and fluid flow in the heart-and how new algorithms aimed at increasing the efficiency of the code may be incorporated.

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 title: ChasteCatalogue identifier: AEFD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFD_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: LGPL 2.1No. of lines in distributed program, including test data, etc.: 5 407 321No. of bytes in distributed program, including test data, etc.: 42 004 554Distribution format: tar.gzProgramming language: C++Operating system: UnixHas the code been vectorised or parallelized?: Yes. Parallelized using MPI.RAM:  <90 Megabytes<90 Megabytes 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, XercesNature 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.

Many problems in Biology and Engineering are governed by anisotropic reaction–diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately capture the propagating wave while avoiding the appearance of spurious oscillations in the wave front. This work develops a family of macro finite elements amenable for solving anisotropic reaction–diffusion equations with stiff reactive terms. The developed elements are incorporated on a semi-implicit algorithm based on operator splitting that includes adaptive time stepping for handling the stiff reactive term. A linear system is solved on each time step to update the transmembrane potential, whereas the remaining ordinary differential equations are solved uncoupled. The method allows solving the linear system on a coarser mesh thanks to the static condensation of the internal degrees of freedom (DOF) of the macroelements while maintaining the accuracy of the finer mesh. The method and algorithm have been implemented in parallel. The accuracy of the method has been tested on two- and three-dimensional examples demonstrating excellent behavior when compared to standard linear elements. The better performance and scalability of different macro finite elements against standard finite elements have been demonstrated in the simulation of a human heart and a heterogeneous two-dimensional problem with reentrant activity. Results have shown a reduction of up to four times in computational cost for the macro finite elements with respect to equivalent (same number of DOF) standard linear finite elements as well as good scalability properties.

has described the software engineering and computational infrastructure that has been set up as part of the Chaste project. Chaste-Cancer Heart And Soft Tissue Environment-is an open source software package that currently has heart and cancer modelling functionality. This software has been written using a programming paradigm imported from the commercial sector, and has resulted in code that has been subject to a far more rigorous testing procedure than is usual in this field. The exceptionally large number of tests written for each small, individual subroutine of this code allows software development to be carried out with confidence that new developments do not inadvertently degrade earlier functionality. This software may be developed for two main purposes-either to add new physiological functionality, or to incorporate a different numerical algorithm for solving the governing equations which allows a given level of accuracy to be attained more efficiently. In this paper, we will focus on the second of these purposes for the field of cardiac electrophysiological modelling. Whiteley (Annals of Biomedical Engineering, vol. 36, pp. 1398-1408) has developed a numerical algorithm for solving the bidomain equations that utilizes the multi-scale nature of the physiology modelled to enhance computational efficiency. When using this algorithm, only quantities that vary on short time-scales or short length-scales are computed to a fine resolution. Other more slowly varying quantities are computed at a lower resolution. Interpolation is then used to transfer information between different resolutions. Using a simple geometry in two dimensions and a purpose-built code, this algorithm was reported to give an increase in computational efficiency of over two orders of magnitude. In this paper, we begin by reviewing numerical methods currently in use for solving the bidomain equations, explaining how these methods may be developed to utilize the multi-scale algorithm discussed above. We then demonstrate the use of this algorithm within the Chaste framework for solving the monodomain and bido- † Author for correspondence (jonathan.whiteley@comlab.ox.ac.uk) main equations in a three-dimensional realistic heart geometry. Finally, we discuss how Chaste may be developed to include new physiological functionality-such as modelling a beating heart and fluid flow in the heart-and how new algorithms aimed at increasing the efficiency of the code may be incorporated.

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.