The Noble cardiac ventricular electrophysiology models in CellML

We present a review of the cardiac ventricular cell electrophysiology models developed by Prof. Denis Noble and colleagues as an example of how models may be published using a web-based CellML publication framework. The models reviewed have been marked-up in CellML and then used to compute all results presented here. The models are freely available from a website as are the specific numerical experiments discussed in this review and the tools used to perform the simulations.     

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.     

Soft Tissue Modelling of Cardiac Fibres for Use in Coupled Mechano-Electric Simulations

The numerical solution of the coupled system of partial differential and ordinary differential equations that model the whole heart in three dimensions is a considerable computational challenge. As a consequence, it is not computationally practical—either in terms of memory or time—to repeat simulations on a finer computational mesh to ensure that convergence of the solution has been attained. In an attempt to avoid this problem while retaining mathematical rigour, we derive a one dimensional model of a cardiac fibre that takes account of elasticity properties in three structurally defined axes within the myocardial tissue. This model of a cardiac fibre is then coupled with an electrophysiological cell model and a model of cellular electromechanics to allow us to simulate the coupling of the electrical and mechanical activity of the heart. We demonstrate that currently used numerical methods for coupling electrical and mechanical activity do not work in this case, and identify appropriate numerical techniques that may be used when solving the governing equations. This allows us to perform a series of simulations that: (i) investigate the effect of some of the assumptions inherent in other models; and (ii) reproduce qualitatively some experimental observations.     

CellML2SBML: conversion of CellML into SBML

CellML and SBML are XML-based languages for storage and exchange of molecular biological and physiological reaction models. They use very similar subsets of MathML to specify the mathematical aspects of the models. CellML2SBML is implemented as a suite of XSLT stylesheets that, when applied consecutively, convert models expressed in CellML into SBML without significant loss of information. The converter is based on the most recent stable versions of the languages (CellML version 1.1; SBML Level 2 Version 1), and the XSLT used in the stylesheets adheres to the XSLT version 1.0 specification. Of all 306 models in the CellML repository in April 2005, CellML2SBML converted 91% automatically into SBML. Minor manual changes to the unit definitions in the originals raised the percentage of successful conversions to 96%. Availability: http://sbml.org/software/cellml2sbml/. Supplementary information: Instructions for use and further documentation available on http://sbml.org/software/cellml2sbml/     

The CellML Model Repository

SUMMARY: The CellML Model Repository provides free access to over 330 biological models. The vast majority of these models are derived from published, peer-reviewed papers. Model curation is an important and ongoing process to ensure the CellML model is able to accurately reproduce the published results. As the CellML community grows, and more people add their models to the repository, model annotation will become increasingly important to facilitate data searches and information retrieval. AVAILABILITY: The CellML Model Repository is publicly accessible at http://www.cellml.org/models.     

Efficient probabilistic model personalization integrating uncertainty on data and parameters: Application to Eikonal-Diffusion models in cardiac electrophysiology

Biophysical models are increasingly used for medical applications at the organ scale. However, model predictions are rarely associated with a confidence measure although there are important sources of uncertainty in computational physiology methods. For instance, the sparsity and noise of the clinical data used to adjust the model parameters (personalization), and the difficulty in modeling accurately soft tissue physiology. The recent theoretical progresses in stochastic models make their use computationally tractable, but there is still a challenge in estimating patient-specific parameters with such models. In this work we propose an efficient Bayesian inference method for model personalization using polynomial chaos and compressed sensing. This method makes Bayesian inference feasible in real 3D modeling problems. We demonstrate our method on cardiac electrophysiology. We first present validation results on synthetic data, then we apply the proposed method to clinical data. We demonstrate how this can help in quantifying the impact of the data characteristics on the personalization (and thus prediction) results. Described method can be beneficial for the clinical use of personalized models as it explicitly takes into account the uncertainties on the data and the model parameters while still enabling simulations that can be used to optimize treatment. Such uncertainty handling can be pivotal for the proper use of modeling as a clinical tool, because there is a crucial requirement to know the confidence one can have in personalized models.     

Towards accurate numerical method for monodomain models using a realistic heart geometry

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

CellML: its future, present and past

Advances in biotechnology and experimental techniques have lead to the elucidation of vast amounts of biological data. Mathematical models provide a method of analysing this data; however, there are two issues that need to be addressed: (1) the need for standards for defining cell models so they can, for example, be exchanged across the World Wide Web, and also read into simulation software in a consistent format and (2) eliminating the errors which arise with the current method of model publication. CellML has evolved to meet these needs of the modelling community. CellML is a free, open-source, eXtensible markup language based standard for defining mathematical models of cellular function. In this paper we summarise the structure of CellML, its current applications (including biological pathway and electrophysiological models), and its future development—in particular, the development of toolsets and the integration of ontologies.     

High-throughput functional curation of cellular electrophysiology models

Effective reuse of a quantitative mathematical model requires not just access to curated versions of the model equations, but also an understanding of the functional capabilities of the model, and the advisable scope of its application. To enable this “functional curation” we have developed a simulation environment that provides high-throughput evaluation of a mathematical model’s functional response to an arbitrary user-defined protocol, and optionally compares the results against experimental data. In this study we demonstrate the efficacy of this simulation environment on 31 cardiac electrophysiology cell models using two test cases. The S1-S2 response is evaluated to characterise the models’ restitution curves, and their L-type calcium channel current-voltage curves are evaluated. The significant variation in the response of these models, even when the models represent the same species and temperature, demonstrates the importance of knowing the functional characteristics of a model prior to its reuse.We also discuss the wider implications for this approach, in improving the selection of models for reuse, enabling the identification of models that exhibit particular experimentally observed phenomena, and making the incremental development of models more robust.     

UML as a cell and biochemistry modeling language

The systems biology community is building increasingly complex models and simulations of cells and other biological entities, and are beginning to look at alternatives to traditional representations such as those provided by ordinary differential equations (ODE). The lessons learned over the years by the software development community in designing and building increasingly complex telecommunication and other commercial real-time reactive systems, can be advantageously applied to the problems of modeling in the biology domain. Making use of the object-oriented (OO) paradigm, the unified modeling language (UML) and Real-Time Object-Oriented Modeling (ROOM) visual formalisms, and the Rational Rose RealTime (RRT) visual modeling tool, we describe a multi-step process we have used to construct top–down models of cells and cell aggregates. The simple example model described in this paper includes membranes with lipid bilayers, multiple compartments including a variable number of mitochondria, substrate molecules, enzymes with reaction rules, and metabolic pathways. We demonstrate the relevance of abstraction, reuse, objects, classes, component and inheritance hierarchies, multiplicity, visual modeling, and other current software development best practices. We show how it is possible to start with a direct diagrammatic representation of a biological structure such as a cell, using terminology familiar to biologists, and by following a process of gradually adding more and more detail, arrive at a system with structure and behavior of arbitrary complexity that can run and be observed on a computer. We discuss our CellAK (Cell Assembly Kit) approach in terms of features found in SBML, CellML, E-CELL, Gepasi, Jarnac, StochSim, Virtual Cell, and membrane computing systems.     

Program Code Generator for Cardiac Electrophysiology Simulation with Automatic PDE Boundary Condition Handling

Clinical and experimental studies involving human hearts can have certain limitations. Methods such as computer simulations can be an important alternative or supplemental tool. Physiological simulation at the tissue or organ level typically involves the handling of partial differential equations (PDEs). Boundary conditions and distributed parameters, such as those used in pharmacokinetics simulation, add to the complexity of the PDE solution. These factors can tailor PDE solutions and their corresponding program code to specific problems. Boundary condition and parameter changes in the customized code are usually prone to errors and time-consuming. We propose a general approach for handling PDEs and boundary conditions in computational models using a replacement scheme for discretization. This study is an extension of a program generator that we introduced in a previous publication. The program generator can generate code for multi-cell simulations of cardiac electrophysiology. Improvements to the system allow it to handle simultaneous equations in the biological function model as well as implicit PDE numerical schemes. The replacement scheme involves substituting all partial differential terms with numerical solution equations. Once the model and boundary equations are discretized with the numerical solution scheme, instances of the equations are generated to undergo dependency analysis. The result of the dependency analysis is then used to generate the program code. The resulting program code are in Java or C programming language. To validate the automatic handling of boundary conditions in the program code generator, we generated simulation code using the FHN, Luo-Rudy 1, and Hund-Rudy cell models and run cell-to-cell coupling and action potential propagation simulations. One of the simulations is based on a published experiment and simulation results are compared with the experimental data. We conclude that the proposed program code generator can be used to generate code for physiological simulations and provides a tool for studying cardiac electrophysiology.     

The Cardiac Electrophysiology Web Lab

Computational modeling of cardiac cellular electrophysiology has a long history, and many models are now available for different species, cell types, and experimental preparations. This success brings with it a challenge: how do we assess and compare the underlying hypotheses and emergent behaviors so that we can choose a model as a suitable basis for a new study or to characterize how a particular model behaves in different scenarios? We have created an online resource for the characterization and comparison of electrophysiological cell models in a wide range of experimental scenarios. The details of the mathematical model (quantitative assumptions and hypotheses formulated as ordinary differential equations) are separated from the experimental protocol being simulated. Each model and protocol is then encoded in computer-readable formats. A simulation tool runs virtual experiments on models encoded in CellML, and a website (https://chaste.cs.ox.ac.uk/WebLab) provides a friendly interface, allowing users to store and compare results. The system currently contains a sample of 36 models and 23 protocols, including current-voltage curve generation, action potential properties under steady pacing at different rates, restitution properties, block of particular channels, and hypo-/hyperkalemia. This resource is publicly available, open source, and free, and we invite the community to use it and become involved in future developments. Investigators interested in comparing competing hypotheses using models can make a more informed decision, and those developing new models can upload them for easy evaluation under the existing protocols, and even add their own protocols.     

Numerical prediction of freezing fronts in cryosurgery: comparison with experimental results

Recent developments in scientific computing now allow to consider realistic applications of numerical modelling to medicine. In this work, a numerical method is presented for the simulation of phase change occurring in cryosurgery applications. The ultimate goal of these simulations is to accurately predict the freezing front position and the thermal history inside the ice ball which is essential to determine if cancerous cells have been completely destroyed. A semi-phase field formulation including blood flow considerations is employed for the simulations. Numerical results are enhanced by the introduction of an anisotropic remeshing strategy. The numerical procedure is validated by comparing the predictions of the model with experimental results.     

On the potentialities of 3D–1D coupled models in hemodynamics simulations

This work comprises a step towards the quantitative and qualitative analysis of coupled local and global hemodynamics phenomena in the arterial system. The aim of this work is to present some numerical examples to put in evidence the importance of the use of 3D–1D coupled models in hemodynamics problems when carrying out simulations of rather complex situations. Accordingly, some cases for which classical modeling cannot be applied are identified and solved. The results obtained here allow us to assess some interrelations between local pointwise quantities (defined at the level of the 3D model) and global mean quantities (defined at the level of the 1D model).