Verification of cardiac tissue electrophysiology simulators using an N-version benchmark

Using physiologically based models for clinical translation: predictive modelling, data interpretation or something in-between?

Heart disease continues to be a significant clinical problem in Western society. Predictive models and simulations that integrate physiological understanding with patient information derived from clinical data have huge potential to contribute to improving our understanding of both the progression and treatment of heart disease. In particular they provide the potential to improve patient selection and optimisation of cardiovascular interventions across a range of pathologies. Currently a significant proportion of this potential is still to be realised. In this paper we discuss the opportunities and challenges associated with this realisation. Reviewing the successful elements of model translation for biophysically based models and the emerging supporting technologies, we propose three distinct modes of clinical translation. Finally we outline the challenges ahead that will be fundamental to overcome if the ultimate goal of fully personalised clinical cardiac care is to be achieved.

Space-discretization error analysis and stabilization schemes for conduction velocity in cardiac electrophysiology

In cardiac electrophysiology, the propagation of the action potential may be described by a set of reaction-diffusion equations known as the bidomain model. The shape of the solution is determined by a balance of a strong reaction and a relatively weak diffusion, which leads to steep variations in space and time. From a numerical point of view, the sharp spatial gradients may be seen as particularly problematic, because computational grid resolution on the order of 0.1 mm or less is required, yielding considerable computational efforts on human geometries. In this paper, we discuss a number of well-known numerical schemes for the bidomain equation and show how the quality of the solution is affected by the spatial discretization. In particular, we study in detail the effect of discretization on the conduction velocity (CV), which is an important quantity from a physiological point of view. We show that commonly applied finite element techniques tend to overestimate the CV on coarse grids, while it tends to be underestimated by finite difference schemes. Furthermore, the choice of interpolation and discretization scheme for the nonlinear reaction term has a strong impact on the CV. Finally, we exploit the results of the error analysis to propose improved numerical methods, including a stabilized scheme that tends to correct the CV on coarse grids but converges to the correct solution as the grid is refined. Copyright © 2016 John Wiley & Sons, Ltd.

Parametrization of activation based cardiac electrophysiology models using bidomain model simulations

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%.

Generating Purkinje networks in the human heart

The Purkinje network is an integral part of the excitation system in the human heart. Yet, to date, there is no in vivo imaging technique to accurately reconstruct its geometry and structure. Computational modeling of the Purkinje network is increasingly recognized as an alternative strategy to visualize, simulate, and understand the role of the Purkinje system. However, most computational models either have to be generated manually, or fail to smoothly cover the irregular surfaces inside the left and right ventricles. Here we present a new algorithm to reliably create robust Purkinje networks within the human heart. We made the source code of this algorithm freely available online. Using Monte Carlo simulations, we demonstrate that the fractal tree algorithm with our new projection method generates denser and more compact Purkinje networks than previous approaches on irregular surfaces. Under similar conditions, our algorithm generates a network with 1219±61 branches, three times more than a conventional algorithm with 419±107 branches. With a coverage of 11±3mm, the surface density of our new Purkije network is twice as dense as the conventional network with 22±7mm. To demonstrate the importance of a dense Purkinje network in cardiac electrophysiology, we simulated three cases of excitation: with our new Purkinje network, with left-sided Purkinje network, and without Purkinje network. Simulations with our new Purkinje network predicted more realistic activation sequences and activation times than simulations without. Six-lead electrocardiograms of the three case studies agreed with the clinical electrocardiograms under physiological conditions, under pathological conditions of right bundle branch block, and under pathological conditions of trifascicular block. Taken together, our results underpin the importance of the Purkinje network in realistic human heart simulations. Human heart modeling has the potential to support the design of personalized strategies for single- or bi-ventricular pacing, radiofrequency ablation, and cardiac defibrillation with the common goal to restore a normal heart rhythm.

Electrophysiology of Heart Failure Using a Rabbit Model: From the Failing Myocyte to Ventricular Fibrillation

Heart failure is a leading cause of death, yet its underlying electrophysiological (EP) mechanisms are not well understood. In this study, we use a multiscale approach to analyze a model of heart failure and connect its results to features of the electrocardiogram (ECG). The heart failure model is derived by modifying a previously validated electrophysiology model for a healthy rabbit heart. Specifically, in accordance with the heart failure literature, we modified the cell EP by changing both membrane currents and calcium handling. At the tissue level, we modeled the increased gap junction lateralization and lower conduction velocity due to downregulation of Connexin 43. At the biventricular level, we reduced the apex-to-base and transmural gradients of action potential duration (APD). The failing cell model was first validated by reproducing the longer action potential, slower and lower calcium transient, and earlier alternans characteristic of heart failure EP. Subsequently, we compared the electrical wave propagation in one dimensional cables of healthy and failing cells. The validated cell model was then used to simulate the EP of heart failure in an anatomically accurate biventricular rabbit model. As pacing cycle length decreases, both the normal and failing heart develop T-wave alternans, but only the failing heart shows QRS alternans (although moderate) at rapid pacing. Moreover, T-wave alternans is significantly more pronounced in the failing heart. At rapid pacing, APD maps show areas of conduction block in the failing heart. Finally, accelerated pacing initiated wave reentry and breakup in the failing heart. Further, the onset of VF was not observed with an upregulation of SERCA, a potential drug therapy, using the same protocol. The changes introduced at the cell and tissue level have increased the failing heart’s susceptibility to dynamic instabilities and arrhythmias under rapid pacing. However, the observed increase in arrhythmogenic potential is not due to a steepening of the restitution curve (not present in our model), but rather to a novel blocking mechanism.

Ventricular fibrillation (VF) is one of the leading causes of sudden death. During VF, the electrical wave of activation in the heart breaks up chaotically. Consequently, the heart is unable to contract synchronously and pump blood to the rest of the body. In our work we formulate and validate a model of heart failure (HF) that allows us to evaluate the arrhythmogenic potential of individual and combined electrophysiological changes. In diagnostic cardiology, the electrocardiogram (ECG) is one of the most commonly used tools for detecting abnormalities in the heart electrophysiology. One of our goals is to use our numerical model to link changes at the cellular and tissue level in a failing heart to a numerically computed ECG. This allows us to characterize the precursor to and the risk of VF. In order to understand the mechanisms underlying VF in HF, we design a test that simulates a HF patient performing physical exercise. We show that under fast heart rates with changes in pacing, HF patients are more prone to VF due to a new conduction blocking mechanism. In the long term, our mathematical model is suitable for investigating the effect of drug therapies in HF.

Cardiac image modelling: Breadth and depth in heart disease

With the advent of large-scale imaging studies and big health data, and the corresponding growth in analytics, machine learning and computational image analysis methods, there are now exciting opportunities for deepening our understanding of the mechanisms and characteristics of heart disease. Two emerging fields are computational analysis of cardiac remodelling (shape and motion changes due to disease) and computational analysis of physiology and mechanics to estimate biophysical properties from non-invasive imaging. Many large cohort studies now underway around the world have been specifically designed based on non-invasive imaging technologies in order to gain new information about the development of heart disease from asymptomatic to clinical manifestations. These give an unprecedented breadth to the quantification of population variation and disease development. Also, for the individual patient, it is now possible to determine biophysical properties of myocardial tissue in health and disease by interpreting detailed imaging data using computational modelling. For these population and patient-specific computational modelling methods to develop further, we need open benchmarks for algorithm comparison and validation, open sharing of data and algorithms, and demonstration of clinical efficacy in patient management and care. The combination of population and patient-specific modelling will give new insights into the mechanisms of cardiac disease, in particular the development of heart failure, congenital heart disease, myocardial infarction, contractile dysfunction and diastolic dysfunction.

Anatomically accurate high resolution modeling of human whole heart electromechanics: A strongly scalable algebraic multigrid solver method for nonlinear deformation

Electromechanical (EM) models of the heart have been used successfully to study fundamental mechanisms underlying a heart beat in health and disease. However, in all modeling studies reported so far numerous simplifications were made in terms of representing biophysical details of cellular function and its heterogeneity, gross anatomy and tissue microstructure, as well as the bidirectional coupling between electrophysiology (EP) and tissue distension. One limiting factor is the employed spatial discretization methods which are not sufficiently flexible to accommodate complex geometries or resolve heterogeneities, but, even more importantly, the limited efficiency of the prevailing solver techniques which are not sufficiently scalable to deal with the incurring increase in degrees of freedom (DOF) when modeling cardiac electromechanics at high spatio-temporal resolution. This study reports on the development of a novel methodology for solving the nonlinear equation of finite elasticity using human whole organ models of cardiac electromechanics, discretized at a high para-cellular resolution. Three patient-specific, anatomically accurate, whole heart EM models were reconstructed from magnetic resonance (MR) scans at resolutions of 220 μm, 440 μm and 880 μm, yielding meshes of approximately 184.6, 24.4 and 3.7 million tetrahedral elements and 95.9, 13.2 and 2.1 million displacement DOF, respectively. The same mesh was used for discretizing the governing equations of both electrophysiology (EP) and nonlinear elasticity. A novel algebraic multigrid (AMG) preconditioner for an iterative Krylov solver was developed to deal with the resulting computational load. The AMG preconditioner was designed under the primary objective of achieving favorable strong scaling characteristics for both setup and solution runtimes, as this is key for exploiting current high performance computing hardware. Benchmark results using the 220 μm, 440 μm and 880 μm meshes demonstrate efficient scaling up to 1024, 4096 and 8192 compute cores which allowed the simulation of a single heart beat in 44.3, 87.8 and 235.3 minutes, respectively. The efficiency of the method allows fast simulation cycles without compromising anatomical or biophysical detail.

Protective Role of False Tendon in Subjects with Left Bundle Branch Block: A Virtual Population Study

False tendons (FTs) are fibrous or fibromuscular bands that can be found in both the normal and abnormal human heart in various anatomical forms depending on their attachment points, tissue types, and geometrical properties. While FTs are widely considered to affect the function of the heart, their specific roles remain largely unclear and unexplored. In this paper, we present an in silico study of the ventricular activation time of the human heart in the presence of FTs. This study presents the first computational model of the human heart that includes a FT, Purkinje network, and papillary muscles. Based on this model, we perform simulations to investigate the effect of different types of FTs on hearts with the electrical conduction abnormality of a left bundle branch block (LBBB). We employ a virtual population of 70 human hearts derived from a statistical atlas, and run a total of 560 simulations to assess ventricular activation time with different FT configurations. The obtained results indicate that, in the presence of a LBBB, the FT reduces the total activation time that is abnormally augmented due to a branch block, to such an extent that surgical implant of cardiac resynchronisation devices might not be recommended by international guidelines. Specifically, the simulation results show that FTs reduce the QRS duration at least 10 ms in 80% of hearts, and up to 45 ms for FTs connecting to the ventricular free wall, suggesting a significant reduction of cardiovascular mortality risk. In further simulation studies we show the reduction in the QRS duration is more sensitive to the shape of the heart then the size of the heart or the exact location of the FT. Finally, the model suggests that FTs may contribute to reducing the activation time difference between the left and right ventricles from 12 ms to 4 ms. We conclude that FTs may provide an alternative conduction pathway that compensates for the propagation delay caused by the LBBB. Further investigation is needed to quantify the clinical impact of FTs on cardiovascular mortality risk.

Image-Based Personalization of Cardiac Anatomy for Coupled Electromechanical Modeling

Computational models of cardiac electromechanics (EM) are increasingly being applied to clinical problems, with patient-specific models being generated from high fidelity imaging and used to simulate patient physiology, pathophysiology and response to treatment. Current structured meshes are limited in their ability to fully represent the detailed anatomical data available from clinical images and capture complex and varied anatomy with limited geometric accuracy. In this paper, we review the state of the art in image-based personalization of cardiac anatomy for biophysically detailed, strongly coupled EM modeling, and present our own tools for the automatic building of anatomically and structurally accurate patient-specific models. Our method relies on using high resolution unstructured meshes for discretizing both physics, electrophysiology and mechanics, in combination with efficient, strongly scalable solvers necessary to deal with the computational load imposed by the large number of degrees of freedom of these meshes. These tools permit automated anatomical model generation and strongly coupled EM simulations at an unprecedented level of anatomical and biophysical detail.

Stability analysis of the POD reduced order method for solving the bidomain model in cardiac electrophysiology

In this paper we show the numerical stability of the Proper Orthogonal Decomposition (POD) reduced order method used in cardiac electrophysiology applications. The difficulty of proving the stability comes from the fact that we are interested in the bidomain model, which is a system of degenerate parabolic equations coupled to a system of ODEs representing the cell membrane electrical activity. The proof of the stability of this method is based on a priori estimates controlling the gap between the reduced order solution and the Galerkin finite element one. We present some numerical simulations confirming the theoretical results. We also combine the POD method with a time splitting scheme allowing a faster solving of the bidomain problem and show numerical results. Finally, we conduct numerical simulation in 2D illustrating the stability of the POD method in its sensitivity to the ionic model parameters. We also perform 3D simulation using a massively parallel code. We show the computational gain using the POD reduced order model. We also show that this method has a better scalability than the full finite element method.

Verification of cardiac mechanics software: benchmark problems and solutions for testing active and passive material behaviour

Models of cardiac mechanics are increasingly used to investigate cardiac physiology. These models are characterized by a high level of complexity, including the particular anisotropic material properties of biological tissue and the actively contracting material. A large number of independent simulation codes have been developed, but a consistent way of verifying the accuracy and replicability of simulations is lacking. To aid in the verification of current and future cardiac mechanics solvers, this study provides three benchmark problems for cardiac mechanics. These benchmark problems test the ability to accurately simulate pressure-type forces that depend on the deformed objects geometry, anisotropic and spatially varying material properties similar to those seen in the left ventricle and active contractile forces. The benchmark was solved by 11 different groups to generate consensus solutions, with typical differences in higher-resolution solutions at approximately 0.5%, and consistent results between linear, quadratic and cubic finite elements as well as different approaches to simulating incompressible materials. Online tools and solutions are made available to allow these tests to be effectively used in verification of future cardiac mechanics software.

Simulating Cardiac Electrophysiology Using Unstructured All-Hexahedra Spectral Elements

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 elements of the proposed approach on some selected benchmarks and on an anatomically based whole heart human computational model.

ECG-Based Detection of Early Myocardial Ischemia in a Computational Model: Impact of Additional Electrodes, Optimal Placement, and a New Feature for ST Deviation

In case of chest pain, immediate diagnosis of myocardial ischemia is required to respond with an appropriate treatment. The diagnostic capability of the electrocardiogram (ECG), however, is strongly limited for ischemic events that do not lead to ST elevation. This computational study investigates the potential of different electrode setups in detecting early ischemia at 10 minutes after onset: standard 3-channel and 12-lead ECG as well as body surface potential maps (BSPMs). Further, it was assessed if an additional ECG electrode with optimized position or the right-sided Wilson leads can improve sensitivity of the standard 12-lead ECG. To this end, a simulation study was performed for 765 different locations and sizes of ischemia in the left ventricle. Improvements by adding a single, subject specifically optimized electrode were similar to those of the BSPM: 2–11% increased detection rate depending on the desired specificity. Adding right-sided Wilson leads had negligible effect. Absence of ST deviation could not be related to specific locations of the ischemic region or its transmurality. As alternative to the ST time integral as a feature of ST deviation, the K point deviation was introduced: the baseline deviation at the minimum of the ST-segment envelope signal, which increased 12-lead detection rate by 7% for a reasonable threshold.

Image-Based Personalization of Cardiac Anatomy for Coupled Electromechanical Modeling

Computational models of cardiac electromechanics (EM) are increasingly being applied to clinical problems, with patient-specific models being generated from high fidelity imaging and used to simulate patient physiology, pathophysiology and response to treatment. Current structured meshes are limited in their ability to fully represent the detailed anatomical data available from clinical images and capture complex and varied anatomy with limited geometric accuracy. In this paper, we review the state of the art in image-based personalization of cardiac anatomy for biophysically detailed, strongly coupled EM modeling, and present our own tools for the automatic building of anatomically and structurally accurate patient-specific models. Our method relies on using high resolution unstructured meshes for discretizing both physics, electrophysiology and mechanics, in combination with efficient, strongly scalable solvers necessary to deal with the computational load imposed by the large number of degrees of freedom of these meshes. These tools permit automated anatomical model generation and strongly coupled EM simulations at an unprecedented level of anatomical and biophysical detail.

Accelerating mono-domain cardiac electrophysiology simulations using OpenCL

Using OpenCL, we developed a cross-platform software to compute electrical excitation conduction in cardiac tissue. OpenCL allowed the software to run parallelized and on different computing devices (e.g., CPUs and GPUs). We used the macroscopic mono-domain model for excitation conduction and an atrial myocyte model by Courtemanche et al. for ionic currents. On a CPU with 12 HyperThreading-enabled Intel Xeon 2.7 GHz cores, we achieved a speed-up of simulations by a factor of 1.6 against existing software that uses OpenMPI. On two high-end AMD FirePro D700 GPUs the OpenCL software ran 2.4 times faster than the OpenMPI implementation. The more nodes the discretized simulation domain contained, the higher speed-ups were achieved.

Models of ventricular arrhythmia mechanisms

The mechanisms that initiate and sustain ventricular arrhythmias in the human heart are clinically important, but hard to study experimentally. In this study, a monodomain model of electrical activation was used to examine how dynamics of electrophysiology at the cell scale influence the surface activation patterns of VF at the tissue scale. Cellular electrophysiology was described with two variants of a phenomenological model of the human ventricular epicardial action potential. The tissue geometry was an 8.0 × 8.0 × 1.2 cm 3D tissue slab with axially symmetric anisotropy. In both cases an initial re-entrant wave fragmented into multiple wavelets of activation. The model variant with steep action potential duration restitution produced much more complex activation, with a greater average number of filaments (13.79) than the variant with less steep restitution (3.08). More complex activation was associated with proportionally fewer transmural filaments, and so the average number of epicardial wavefronts and phase singularities per filament was lower. The average number of epicardial phase singularities and wavefronts for the model variant with less steep restitution were consistent with experimental observations in the human heart. This study shows that small changes in cell scale dynamics can have a large influence on the complexity of re-entrant activation in simulated 3D tissue, as well as on the features observed on the epicardial surface.

High-order finite element methods for cardiac monodomain simulations

Computational modeling of tissue-scale cardiac electrophysiology requires numerically converged solutions to avoid spurious artifacts. The steep gradients inherent to cardiac action potential propagation necessitate fine spatial scales and therefore a substantial computational burden. The use of high-order interpolation methods has previously been proposed for these simulations due to their theoretical convergence advantage. In this study, we compare the convergence behavior of linear Lagrange, cubic Hermite, and the newly proposed cubic Hermite-style serendipity interpolation methods for finite element simulations of the cardiac monodomain equation. The high-order methods reach converged solutions with fewer degrees of freedom and longer element edge lengths than traditional linear elements. Additionally, we propose a dimensionless number, the cell Thiele modulus, as a more useful metric for determining solution convergence than element size alone. Finally, we use the cell Thiele modulus to examine convergence criteria for obtaining clinically useful activation patterns for applications such as patient-specific modeling where the total activation time is known a priori.

Three-dimensional histology: tools and application to quantitative assessment of cell-type distribution in rabbit heart

Aims

Cardiac histo-anatomical organization is a major determinant of function. Changes in tissue structure are a relevant factor in normal and disease development, and form targets of therapeutic interventions. The purpose of this study was to test tools aimed to allow quantitative assessment of cell-type distribution from large histology and magnetic resonance imaging- (MRI) based datasets.

Methods and results

Rabbit heart fixation during cardioplegic arrest and MRI were followed by serial sectioning of the whole heart and light-microscopic imaging of trichrome-stained tissue. Segmentation techniques developed specifically for this project were applied to segment myocardial tissue in the MRI and histology datasets. In addition, histology slices were segmented into myocytes, connective tissue, and undefined. A bounding surface, containing the whole heart, was established for both MRI and histology. Volumes contained in the bounding surface (called ‘anatomical volume’), as well as that identified as containing any of the above tissue categories (called ‘morphological volume’), were calculated. The anatomical volume was 7.8 cm³ in MRI, and this reduced to 4.9 cm³ after histological processing, representing an ‘anatomical’ shrinkage by 37.2%. The morphological volume decreased by 48% between MRI and histology, highlighting the presence of additional tissue-level shrinkage (e.g. an increase in interstitial cleft space). The ratio of pixels classified as containing myocytes to pixels identified as non-myocytes was roughly 6:1 (61.6 vs. 9.8%; the remaining fraction of 28.6% was ‘undefined’).

Conclusion

Qualitative and quantitative differentiation between myocytes and connective tissue, using state-of-the-art high-resolution serial histology techniques, allows identification of cell-type distribution in whole-heart datasets. Comparison with MRI illustrates a pronounced reduction in anatomical and morphological volumes during histology processing.

Three-dimensional cardiac computational modelling: methods, features and applications

The combination of computational models and biophysical simulations can help to interpret an array of experimental data and contribute to the understanding, diagnosis and treatment of complex diseases such as cardiac arrhythmias. For this reason, three-dimensional (3D) cardiac computational modelling is currently a rising field of research. The advance of medical imaging technology over the last decades has allowed the evolution from generic to patient-specific 3D cardiac models that faithfully represent the anatomy and different cardiac features of a given alive subject. Here we analyse sixty representative 3D cardiac computational models developed and published during the last fifty years, describing their information sources, features, development methods and online availability. This paper also reviews the necessary components to build a 3D computational model of the heart aimed at biophysical simulation, paying especial attention to cardiac electrophysiology (EP), and the existing approaches to incorporate those components. We assess the challenges associated to the different steps of the building process, from the processing of raw clinical or biological data to the final application, including image segmentation, inclusion of substructures and meshing among others. We briefly outline the personalisation approaches that are currently available in 3D cardiac computational modelling. Finally, we present examples of several specific applications, mainly related to cardiac EP simulation and model-based image analysis, showing the potential usefulness of 3D cardiac computational modelling into clinical environments as a tool to aid in the prevention, diagnosis and treatment of cardiac diseases.

Increased cell membrane capacitance is the dominant mechanism of stretch-dependent conduction slowing in the rabbit heart: a computational study

Volume loading of the cardiac ventricles is known to slow electrical conduction in the rabbit heart, but the mechanisms remain unclear. Previous experimental and modeling studies have investigated some of these mechanisms, including stretch-activated membrane currents, reduced gap junctional conductance, and altered cell membrane capacitance. In order to quantify the relative contributions of these mechanisms, we combined a monomain model of rabbit ventricular electrophysiology with a hyperelastic model of passive ventricular mechanics. First, a simplified geometric model with prescribed homogeneous deformation was used to fit model parameters and characterize individual MEF mechanisms, and showed good qualitative agreement with experimentally measured strain-CV relations. A 3D model of the rabbit left and right ventricles was then compared with experimental measurements from optical electrical mapping studies in the isolated rabbit heart. The model was inflated to an end-diastolic pressure of 30 mmHg, resulting in epicardial strains comparable to those measured in the anterior left ventricular free wall. While the effects of stretch activated channels did alter epicardial conduction velocity, an increase in cellular capacitance was required to explain previously reported experimental results. The new results suggest that for large strains, various mechanisms can combine and produce a biphasic relationship between strain and conduction velocity. However, at the moderate strains generated by high end-diastolic pressure, a stretch-induced increase in myocyte membrane capacitance is the dominant driver of conduction slowing during ventricular volume loading.

A brief history of tissue models for cardiac electrophysiology

The last four decades have produced a number of significant advances in the developments of computer models to simulate and investigate the electrical activity of cardiac tissue. The tissue descriptions that underlie these simulations have been built from a combination of clever insight and careful comparison with measured data at multiple scales. Tissue models have not only led to greater insights into the mechanisms of life-threatening arrhythmias but have been used to engineer new therapies to treat the consequences of cardiac disease. This paper is a look back at the early years in the cardiac modeling and the challenges facing the field as models move toward the clinic.

Computational cardiac electrophysiology: implementing mathematical models of cardiomyocytes to simulate action potentials of the heart

Mathematical models are now an important tool for studying cardiac electrophysiology and arrhythmias. Utilizing these models to quantify behavior and make predictions requires solving the models computationally using numerical schemes. We discuss different solution methods and other computational considerations for simulating cardiac action potentials in single cells and multicellular preparations.

Simulation Methods and Validation Criteria for Modeling Cardiac Ventricular Electrophysiology

We describe a sequence of methods to produce a partial differential equation model of the electrical activation of the ventricles. In our framework, we incorporate the anatomy and cardiac microstructure obtained from magnetic resonance imaging and diffusion tensor imaging of a New Zealand White rabbit, the Purkinje structure and the Purkinje-muscle junctions, and an electrophysiologically accurate model of the ventricular myocytes and tissue, which includes transmural and apex-to-base gradients of action potential characteristics. We solve the electrophysiology governing equations using the finite element method and compute both a 6-lead precordial electrocardiogram (ECG) and the activation wavefronts over time. We are particularly concerned with the validation of the various methods used in our model and, in this regard, propose a series of validation criteria that we consider essential. These include producing a physiologically accurate ECG, a correct ventricular activation sequence, and the inducibility of ventricular fibrillation. Among other components, we conclude that a Purkinje geometry with a high density of Purkinje muscle junctions covering the right and left ventricular endocardial surfaces as well as transmural and apex-to-base gradients in action potential characteristics are necessary to produce ECGs and time activation plots that agree with physiological observations.

A call for virtual experiments: accelerating the scientific process

Experimentation is fundamental to the scientific method, whether for exploration, description or explanation. We argue that promoting the reuse of virtual experiments (the in silico analogues of wet-lab or field experiments) would vastly improve the usefulness and relevance of computational models, encouraging critical scrutiny of models and serving as a common language between modellers and experimentalists. We review the benefits of reusable virtual experiments: in specifying, assaying, and comparing the behavioural repertoires of models; as prerequisites for reproducible research; to guide model reuse and composition; and for quality assurance in the translational application of models. A key step towards achieving this is that models and experimental protocols should be represented separately, but annotated so as to facilitate the linking of models to experiments and data. Lastly, we outline how the rigorous, streamlined confrontation between experimental datasets and candidate models would enable a "continuous integration" of biological knowledge, transforming our approach to systems biology.

Verification of computational models of cardiac electro-physiology

For computational models of cardiac activity to be used in safety-critical clinical decision-making, thorough and rigorous testing of the accuracy of predictions is required. The field of 'verification, validation and uncertainty quantification' has been developed to evaluate the credibility of computational predictions. The first stage, verification, is the evaluation of how well computational software correctly solves the underlying mathematical equations. The aim of this paper is to introduce novel methods for verifying multi-cellular electro-physiological solvers, a crucial first stage for solvers to be used with confidence in clinical applications. We define 1D-3D model problems with exact solutions for each of the monodomain, bidomain, and bidomain-with-perfusing-bath formulations of cardiac electro-physiology, which allow for the first time the testing of cardiac solvers against exact errors on fully coupled problems in all dimensions. These problems are carefully constructed so that they can be easily run using a general solver and can be used to greatly increase confidence that an implementation is correct, which we illustrate by testing one major solver, 'Chaste', on the problems. We then perform case studies on calculation verification (also known as solution verification) for two specific applications. We conclude by making several recommendations regarding verification in cardiac modelling.

Toward GPGPU accelerated human electromechanical cardiac simulations

In this paper, we look at the acceleration of weakly coupled electromechanics using the graphics processing unit (GPU). Specifically, we port to the GPU a number of components of CHeart--a CPU-based finite element code developed for simulating multi-physics problems. On the basis of a criterion of computational cost, we implemented on the GPU the ODE and PDE solution steps for the electrophysiology problem and the Jacobian and residual evaluation for the mechanics problem. Performance of the GPU implementation is then compared with single core CPU (SC) execution as well as multi-core CPU (MC) computations with equivalent theoretical performance. Results show that for a human scale left ventricle mesh, GPU acceleration of the electrophysiology problem provided speedups of 164 × compared with SC and 5.5 times compared with MC for the solution of the ODE model. Speedup of up to 72 × compared with SC and 2.6 × compared with MC was also observed for the PDE solve. Using the same human geometry, the GPU implementation of mechanics residual/Jacobian computation provided speedups of up to 44 × compared with SC and 2.0 × compared with MC.

Ensuring reliability of safety-critical clinical applications of computational cardiac models

Computational models of cardiac electrophysiology have been used for over half a century to investigate physiological mechanisms and generate hypotheses for experimental testing, and are now starting to play a role in clinical applications. There is currently a great deal of interest in using models as diagnostic or therapeutic aids, for example using patient-specific whole-heart simulations to optimize cardiac resynchronization therapy, ablation therapy, and defibrillation. However, if models are to be used in safety-critical clinical decision making, the reliability of their predictions needs to be thoroughly investigated. In engineering and the physical sciences, the field of “verification, validation and uncertainty quantification” (VVUQ) [also known as “verification and validation” (V&V)] has been developed for rigorously evaluating the credibility of computational model predictions. In this article we first discuss why it is vital that cardiac models be developed and evaluated within a VVUQ framework, and then consider cardiac models in the context of each of the stages in VVUQ. We identify some of the major difficulties which may need to be overcome for cardiac models to be used in safely-critical clinical applications.

Numerical quadrature and operator splitting in finite element methods for cardiac electrophysiology

We study the numerical accuracy and computational efficiency of alternative formulations of the finite element solution procedure for the monodomain equations of cardiac electrophysiology, focusing on the interaction of spatial quadrature implementations with operator splitting and examining both nodal and Gauss quadrature methods and implementations that mix nodal storage of state variables with Gauss quadrature. We evaluate the performance of all possible combinations of 'lumped' approximations of consistent capacitance and mass matrices. Most generally, we find that quadrature schemes and lumped approximations that produce decoupled nodal ionic equations allow for the greatest computational efficiency, this being afforded through the use of asynchronous adaptive time-stepping of the ionic state variable ODEs. We identify two lumped approximation schemes that exhibit superior accuracy, rivaling that of the most expensive variationally consistent implementations. Finally, we illustrate some of the physiological consequences of discretization error in electrophysiological simulation relevant to cardiac arrhythmia and fibrillation. These results suggest caution with the use of semi-automated free-form tetrahedral and hexahedral meshing algorithms available in most commercially available meshing software, which produce nonuniform meshes having a large distribution of element sizes.

Computational modeling of chemo-electro-mechanical coupling: a novel implicit monolithic finite element approach

Computational modeling of the human heart allows us to predict how chemical, electrical, and mechanical fields interact throughout a cardiac cycle. Pharmacological treatment of cardiac disease has advanced significantly over the past decades, yet it remains unclear how the local biochemistry of an individual heart cell translates into global cardiac function. Here, we propose a novel, unified strategy to simulate excitable biological systems across three biological scales. To discretize the governing chemical, electrical, and mechanical equations in space, we propose a monolithic finite element scheme. We apply a highly efficient and inherently modular global-local split, in which the deformation and the transmembrane potential are introduced globally as nodal degrees of freedom, whereas the chemical state variables are treated locally as internal variables. To ensure unconditional algorithmic stability, we apply an implicit backward Euler finite difference scheme to discretize the resulting system in time. To increase algorithmic robustness and guarantee optimal quadratic convergence, we suggest an incremental iterative Newton-Raphson scheme. The proposed algorithm allows us to simulate the interaction of chemical, electrical, and mechanical fields during a representative cardiac cycle on a patient-specific geometry, robust and stable, with calculation times on the order of 4 days on a standard desktop computer.

Computational assessment of drug-induced effects on the electrocardiogram: from ion channel to body surface potentials

Background and Purpose

Understanding drug effects on the heart is key to safety pharmacology assessment and anti-arrhythmic therapy development. Here our goal is to demonstrate the ability of computational models to simulate the effect of drug action on the electrical activity of the heart, at the level of the ion-channel, cell, heart and ECG body surface potential.

Experimental Approach

We use the state-of-the-art mathematical models governing the electrical activity of the heart. A drug model is introduced using an ion channel conductance block for the hERG and fast sodium channels, depending on the IC₅₀ value and the drug dose. We simulate the ECG measurements at the body surface and compare biomarkers under different drug actions.

Key Results

Introducing a 50% hERG-channel current block results in 8% prolongation of the APD₉₀ and 6% QT interval prolongation, hERG block does not affect the QRS interval. Introducing 50% fast sodium current block prolongs the QRS and the QT intervals by 12% and 5% respectively, and delays activation times, whereas APD₉₀ is not affected.

Conclusions and Implications

Both potassium and sodium blocks prolong the QT interval, but the underlying mechanism is different: for potassium it is due to APD prolongation; while for sodium it is due to a reduction of electrical wave velocity. This study shows the applicability of in silico models for the investigation of drug effects on the heart, from the ion channel to the ECG-based biomarkers.

Towards real-time simulation of cardiac electrophysiology in a human heart at high resolution

We have developed the capability to rapidly simulate cardiac electrophysiological phenomena in a human heart discretised at a resolution comparable with the length of a cardiac myocyte. Previous scientific investigation has generally invoked simplified geometries or coarse-resolution hearts, with simulation duration limited to 10s of heartbeats. Using state-of-the-art high-performance computing techniques coupled with one of the most powerful computers available (the 20 PFlop/s IBM BlueGene/Q at Lawrence Livermore National Laboratory), high-resolution simulation of the human heart can now be carried out over 1200 times faster compared with published results in the field. We demonstrate the utility of this capability by simulating, for the first time, the formation of transmural re-entrant waves in a 3D human heart. Such wave patterns are thought to underlie Torsades de Pointes, an arrhythmia that indicates a high risk of sudden cardiac death. Our new simulation capability has the potential to impact a multitude of applications in medicine, pharmaceuticals and implantable devices.

Role of temperature on nonlinear cardiac dynamics

Thermal effects affecting spatiotemporal behavior of cardiac tissue are discussed by relating temperature variations to proarrhythmic dynamics in the heart. By introducing a thermoelectric coupling in a minimal model of cardiac tissue, we are able to reproduce experimentally measured dynamics obtained simultaneously from epicardial and endocardial canine right ventricles at different temperatures. A quantitative description of emergent proarrhythmic properties of restitution, conduction velocity, and alternans regimes as a function of temperature is presented. Complex discordant alternans patterns that enhance tissue dispersion consisting of one wave front and three wave backs are described in both simulations and experiments. Possible implications for model generalization are finally discussed.

Combining wet and dry research: experience with model development for cardiac mechano-electric structure-function studies

Since the development of the first mathematical cardiac cell model 50 years ago, computational modelling has become an increasingly powerful tool for the analysis of data and for the integration of information related to complex cardiac behaviour. Current models build on decades of iteration between experiment and theory, representing a collective understanding of cardiac function. All models, whether computational, experimental, or conceptual, are simplified representations of reality and, like tools in a toolbox, suitable for specific applications. Their range of applicability can be explored (and expanded) by iterative combination of 'wet' and 'dry' investigation, where experimental or clinical data are used to first build and then validate computational models (allowing integration of previous findings, quantitative assessment of conceptual models, and projection across relevant spatial and temporal scales), while computational simulations are utilized for plausibility assessment, hypotheses-generation, and prediction (thereby defining further experimental research targets). When implemented effectively, this combined wet/dry research approach can support the development of a more complete and cohesive understanding of integrated biological function. This review illustrates the utility of such an approach, based on recent examples of multi-scale studies of cardiac structure and mechano-electric function.

Chaste: an open source C++ library for computational physiology and biology

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, cell-based 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 test-driven 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 http://www.cs.ox.ac.uk/chaste, together with details of a mailing list and links to documentation and tutorials.

Share and enjoy: anatomical models database--generating and sharing cardiovascular model data using web services

Sharing data between scientists and with clinicians in cardiac research has been facilitated significantly by the use of web technologies. The potential of this technology has meant that information sharing has been routinely promoted through databases that have encouraged stakeholder participation in communities around these services. In this paper we discuss the Anatomical Model Database (AMDB) (Gianni et al. Functional imaging and modeling of the heart. Springer, Heidelberg, 2009; Gianni et al. Phil Trans Ser A Math Phys Eng Sci 368:3039-3056, 2010) which both facilitate a database-centric approach to collaboration, and also extends this framework with new capabilities for creating new mesh data. AMDB currently stores cardiac geometric models described in Gianni et al. (Functional imaging and modelling of the heart. Springer, Heidelberg, 2009), a number of additional cardiac models describing geometry and functional properties, and most recently models generated using a web service. The functional models represent data from simulations in geometric form, such as electrophysiology or mechanics, many of which are present in AMDB as part of a benchmark study. Finally, the heartgen service has been added for producing left or bi-ventricle models derived from binary image data using the methods described in Lamata et al. (Med Image Anal 15:801-813, 2011). The results can optionally be hosted on AMDB alongside other community-provided anatomical models. AMDB is, therefore, a unique database storing geometric data (rather than abstract models or image data) combined with a powerful web service for generating new geometric models.

A three-dimensional human atrial model with fiber orientation. Electrograms and arrhythmic activation patterns relationship

The most common sustained cardiac arrhythmias in humans are atrial tachyarrhythmias, mainly atrial fibrillation. Areas of complex fractionated atrial electrograms and high dominant frequency have been proposed as critical regions for maintaining atrial fibrillation; however, there is a paucity of data on the relationship between the characteristics of electrograms and the propagation pattern underlying them. In this study, a realistic 3D computer model of the human atria has been developed to investigate this relationship. The model includes a realistic geometry with fiber orientation, anisotropic conductivity and electrophysiological heterogeneity. We simulated different tachyarrhythmic episodes applying both transient and continuous ectopic activity. Electrograms and their dominant frequency and organization index values were calculated over the entire atrial surface. Our simulations show electrograms with simple potentials, with little or no cycle length variations, narrow frequency peaks and high organization index values during stable and regular activity as the observed in atrial flutter, atrial tachycardia (except in areas of conduction block) and in areas closer to ectopic activity during focal atrial fibrillation. By contrast, cycle length variations and polymorphic electrograms with single, double and fragmented potentials were observed in areas of irregular and unstable activity during atrial fibrillation episodes. Our results also show: (1) electrograms with potentials without negative deflection related to spiral or curved wavefronts that pass over the recording point and move away, (2) potentials with a much greater proportion of positive deflection than negative in areas of wave collisions, (3) double potentials related with wave fragmentations or blocking lines and (4) fragmented electrograms associated with pivot points. Our model is the first human atrial model with realistic fiber orientation used to investigate the relationship between different atrial arrhythmic propagation patterns and the electrograms observed at more than 43000 points on the atrial surface.

Benchmarking electrophysiological models of human atrial myocytes

Mathematical modeling of cardiac electrophysiology is an insightful method to investigate the underlying mechanisms responsible for arrhythmias such as atrial fibrillation (AF). In past years, five models of human atrial electrophysiology with different formulations of ionic currents, and consequently diverging properties, have been published. The aim of this work is to give an overview of strengths and weaknesses of these models depending on the purpose and the general requirements of simulations. Therefore, these models were systematically benchmarked with respect to general mathematical properties and their ability to reproduce certain electrophysiological phenomena, such as action potential (AP) alternans. To assess the models' ability to replicate modified properties of human myocytes and tissue in cardiac disease, electrical remodeling in chronic atrial fibrillation (cAF) was chosen as test case. The healthy and remodeled model variants were compared with experimental results in single-cell, 1D and 2D tissue simulations to investigate AP and restitution properties, as well as the initiation of reentrant circuits.

Personalization of atrial anatomy and electrophysiology as a basis for clinical modeling of radio-frequency ablation of atrial fibrillation

Multiscale cardiac modeling has made great advances over the last decade. Highly detailed atrial models were created and used for the investigation of initiation and perpetuation of atrial fibrillation. The next challenge is the use of personalized atrial models in clinical practice. In this study, a framework of simple and robust tools is presented, which enables the generation and validation of patient-specific anatomical and electrophysiological atrial models. Introduction of rule-based atrial fiber orientation produced a realistic excitation sequence and a better correlation to the measured electrocardiograms. Personalization of the global conduction velocity lead to a precise match of the measured P-wave duration. The use of a virtual cohort of nine patient and volunteer models averaged out possible model-specific errors. Intra-atrial excitation conduction was personalized manually from left atrial local activation time maps. Inclusion of LE-MRI data into the simulations revealed possible gaps in ablation lesions. A fast marching level set approach to compute atrial depolarization was extended to incorporate anisotropy and conduction velocity heterogeneities and reproduced the monodomain solution. The presented chain of tools is an important step towards the use of atrial models for the patient-specific AF diagnosis and ablation therapy planing.

Application of cardiac electrophysiology simulations to pro-arrhythmic safety testing

Concerns over cardiac side effects are the largest single cause of compound attrition during pharmaceutical drug development. For a number of years, biophysically detailed mathematical models of cardiac electrical activity have been used to explore how a compound, interfering with specific ion-channel function, may explain effects at the cell-, tissue- and organ-scales. With the advent of high-throughput screening of multiple ion channels in the wet-lab, and improvements in computational modelling of their effects on cardiac cell activity, more reliable prediction of pro-arrhythmic risk is becoming possible at the earliest stages of drug development. In this paper, we review the current use of biophysically detailed mathematical models of cardiac myocyte electrical activity in drug safety testing, and suggest future directions to employ the full potential of this approach.

Computational modelling of cardiac electrophysiology: explanation of the variability of results from different numerical solvers

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.

Efficient simulation of cardiac electrical propagation using high order finite elements

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.

Construction of 3D MR image-based computer models of pathologic hearts, augmented with histology and optical fluorescence imaging to characterize action potential propagation

Cardiac computer models can help us understand and predict the propagation of excitation waves (i.e., action potential, AP) in healthy and pathologic hearts. Our broad aim is to develop accurate 3D MR image-based computer models of electrophysiology in large hearts (translatable to clinical applications) and to validate them experimentally. The specific goals of this paper were to match models with maps of the propagation of optical AP on the epicardial surface using large porcine hearts with scars, estimating several parameters relevant to macroscopic reaction-diffusion electrophysiological models. We used voltage-sensitive dyes to image AP in large porcine hearts with scars (three specimens had chronic myocardial infarct, and three had radiofrequency RF acute scars). We first analyzed the main AP waves' characteristics: duration (APD) and propagation under controlled pacing locations and frequencies as recorded from 2D optical images. We further built 3D MR image-based computer models that have information derived from the optical measures, as well as morphologic MRI data (i.e., myocardial anatomy, fiber directions and scar definition). The scar morphology from MR images was validated against corresponding whole-mount histology. We also compared the measured 3D isochronal maps of depolarization to simulated isochrones (the latter replicating precisely the experimental conditions), performing model customization and 3D volumetric adjustments of the local conductivity. Our results demonstrated that mean APD in the border zone (BZ) of the infarct scars was reduced by ~13% (compared to ~318 ms measured in normal zone, NZ), but APD did not change significantly in the thin BZ of the ablation scars. A generic value for velocity ratio (1:2.7) in healthy myocardial tissue was derived from measured values of transverse and longitudinal conduction velocities relative to fibers direction (22 cm/s and 60 cm/s, respectively). The model customization and 3D volumetric adjustment reduced the differences between measurements and simulations; for example, from one pacing location, the adjustment reduced the absolute error in local depolarization times by a factor of 5 (i.e., from 58 ms to 11 ms) in the infarcted heart, and by a factor of 6 (i.e., from 60 ms to 9 ms) in the heart with the RF scar. Moreover, the sensitivity of adjusted conductivity maps to different pacing locations was tested, and the errors in activation times were found to be of approximately 10-12 ms independent of pacing location used to adjust model parameters, suggesting that any location can be used for model predictions.

Teaching cardiac electrophysiology modeling to undergraduate students: laboratory exercises and GPU programming for the study of arrhythmias and spiral wave dynamics

As part of a 3-wk intersession workshop funded by a National Science Foundation Expeditions in Computing award, 15 undergraduate students from the City University of New York(1) collaborated on a study aimed at characterizing the voltage dynamics and arrhythmogenic behavior of cardiac cells for a broad range of physiologically relevant conditions using an in silico model. The primary goal of the workshop was to cultivate student interest in computational modeling and analysis of complex systems by introducing them through lectures and laboratory activities to current research in cardiac modeling and by engaging them in a hands-on research experience. The success of the workshop lay in the exposure of the students to active researchers and experts in their fields, the use of hands-on activities to communicate important concepts, active engagement of the students in research, and explanations of the significance of results as the students generated them. The workshop content addressed how spiral waves of electrical activity are initiated in the heart and how different parameter values affect the dynamics of these reentrant waves. Spiral waves are clinically associated with tachycardia, when the waves remain stable, and with fibrillation, when the waves exhibit breakup. All in silico experiments were conducted by simulating a mathematical model of cardiac cells on graphics processing units instead of the standard central processing units of desktop computers. This approach decreased the run time for each simulation to almost real time, thereby allowing the students to quickly analyze and characterize the simulated arrhythmias. Results from these simulations, as well as some of the background and methodology taught during the workshop, is presented in this article along with the programming code and the explanations of simulation results in an effort to allow other teachers and students to perform their own demonstrations, simulations, and studies.

Effects of boundaries and geometry on the spatial distribution of action potential duration in cardiac tissue

Increased dispersion of action potential duration across cardiac tissue has long been considered an important substrate for the development of most electrical arrhythmias. Although this dispersion has been studied previously by characterizing the static intrinsic gradients in cellular electrophysiology and dynamical gradients generated by fast pacing, few studies have concentrated on dispersions generated solely by structural effects. Here we show how boundaries and geometry can produce spatially dependent changes in action potential duration (APD) in homogeneous and isotropic tissue, where all the cells have the same APD in the absence of diffusion. Electrotonic currents due to coupling within the tissue and at the tissue boundaries can generate dispersion, and the profile of this dispersion can change dramatically depending on tissue size and shape, action potential morphology, tissue dimensionality, and stimulus frequency and location. The dispersion generated by pure geometrical effects can be on the order of tens of milliseconds, enough under certain conditions to produce conduction blocks and initiate reentrant waves.

Considerations for the use of cellular electrophysiology models within cardiac tissue simulations

The use of mathematical models to study cardiac electrophysiology has a long history, and numerous cellular scale models are now available, covering a range of species and cell types. Their use to study emergent properties in tissue is also widespread, typically using the monodomain or bidomain equations coupled to one or more cell models. Despite the relative maturity of this field, little has been written looking in detail at the interface between the cellular and tissue-level models. Mathematically this is relatively straightforward and well-defined. There are however many details and potential inconsistencies that need to be addressed, in order to ensure correct operation of a cellular model within a tissue simulation. This paper will describe these issues and how to address them. Simply having models available in a common format such as CellML is still of limited utility, with significant manual effort being required to integrate these models within a tissue simulation. We will thus also discuss the facilities available for automating this in a consistent fashion within Chaste, our robust and high-performance cardiac electrophysiology simulator. It will be seen that a common theme arising is the need to go beyond a representation of the model mathematics in a standard language, to include additional semantic information required in determining the model's interface, and hence to enhance interoperability. Such information can be added as metadata, but agreement is needed on the terms to use, including development of appropriate ontologies, if reliable automated use of CellML models is to become common.