Rafel Bordas

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.

A numerical guide to the solution of the bidomain equations of cardiac electrophysiology

Simulation of cardiac electrical activity using the bi-domain equations can be a massively computationally demanding problem. This study provides a comprehensive guide to numerical bi-domain modelling. Each component of bi-domain simulations--discretization, ODE-solution, linear system solution, and parallelization--is discussed, and previously-used methods are reviewed, new methods are proposed, and issues which cause particular difficulty are highlighted. Particular attention is paid to the choice of stimulus currents, compatibility conditions for the equations, the solution of singular linear systems, and convergence of the numerical scheme.

Rabbit-specific ventricular model of cardiac electrophysiological function including specialized conduction system.

The function of the ventricular specialized conduction system in the heart is to ensure the coordinated electrical activation of the ventricles. It is therefore critical to the overall function of the heart, and has also been implicated as an important player in various diseases, including lethal ventricular arrhythmias such as ventricular fibrillation and drug-induced torsades de pointes. However, current ventricular models of electrophysiology usually ignore, or include highly simplified representations of the specialized conduction system. Here, we describe the development of an image-based, species-consistent, anatomically-detailed model of rabbit ventricular electrophysiology that incorporates a detailed description of the free-running part of the specialized conduction system. Techniques used for the construction of the geometrical model of the specialized conduction system from a magnetic resonance dataset and integration of the system model into a ventricular anatomical model, developed from the same dataset, are described. Computer simulations of rabbit ventricular electrophysiology are conducted using the novel anatomical model and rabbit-specific membrane kinetics to investigate the importance of the components and properties of the conduction system in determining ventricular function under physiological conditions. Simulation results are compared to panoramic optical mapping experiments for model validation and results interpretation. Full access is provided to the anatomical models developed in this study.

Simulation of cardiac electrophysiology on next-generation high-performance computers

Models of cardiac electrophysiology consist of a system of partial differential equations (PDEs) coupled with a system of ordinary differential equations representing cell membrane dynamics. Current software to solve such models does not provide the required computational speed for practical applications. One reason for this is that little use is made of recent developments in adaptive numerical algorithms for solving systems of PDEs. Studies have suggested that a speedup of up to two orders of magnitude is possible by using adaptive methods. The challenge lies in the efficient implementation of adaptive algorithms on massively parallel computers. The finite-element (FE) method is often used in heart simulators as it can encapsulate the complex geometry and small-scale details of the human heart. An alternative is the spectral element (SE) method, a high-order technique that provides the flexibility and accuracy of FE, but with a reduced number of degrees of freedom. The feasibility of implementing a parallel SE algorithm based on fully unstructured all-hexahedra meshes is discussed. A major computational task is solution of the large algebraic system resulting from FE or SE discretization. Choice of linear solver and preconditioner has a substantial effect on efficiency. A fully parallel implementation based on dynamic partitioning that accounts for load balance, communication and data movement costs is required. Each of these methods must be implemented on next-generation supercomputers in order to realize the necessary speedup. The problems that this may cause, and some of the techniques that are beginning to be developed to overcome these issues, are described.

CHASTE: incorporating a novel multi-scale spatial and temporal algorithm into a large-scale open source library

Recent work has described the software engineering and computational infrastructure that has been set up as part of the Cancer, Heart and Soft Tissue Environment (Chaste) project. Chaste 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 a code that has been subject to a far more rigorous testing procedure than that is usual in this field. In this paper, we explain how new functionality may be incorporated into Chaste. Whiteley has developed a numerical algorithm for solving the bidomain equations that uses the multi-scale (MS) nature of the physiology modelled to enhance computational efficiency. Using a simple geometry in two dimensions and a purpose-built code, this algorithm was reported to give an increase in computational efficiency of more than 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 use the MS algorithm discussed above. We then demonstrate the use of this algorithm within the Chaste framework for solving the monodomain and bidomain 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.

A comparison of fully automated methods of data analysis and computer assisted heuristic methods in an electrode kinetic study of the pathologically variable [Fe(CN)(6)](3-/4-) process by AC voltammetry

Fully automated and computer assisted heuristic data analysis approaches have been applied to a series of AC voltammetric experiments undertaken on the [Fe(CN)6](3-/4-) process at a glassy carbon electrode in 3 M KCl aqueous electrolyte. The recovered parameters in all forms of data analysis encompass E(0) (reversible potential), k(0) (heterogeneous charge transfer rate constant at E(0)), α (charge transfer coefficient), Ru (uncompensated resistance), and Cdl (double layer capacitance). The automated method of analysis employed time domain optimization and Bayesian statistics. This and all other methods assumed the Butler-Volmer model applies for electron transfer kinetics, planar diffusion for mass transport, Ohms Law for Ru, and a potential-independent Cdl model. Heuristic approaches utilize combinations of Fourier Transform filtering, sensitivity analysis, and simplex-based forms of optimization applied to resolved AC harmonics and rely on experimenter experience to assist in experiment-theory comparisons. Remarkable consistency of parameter evaluation was achieved, although the fully automated time domain method provided consistently higher α values than those based on frequency domain data analysis. The origin of this difference is that the implemented fully automated method requires a perfect model for the double layer capacitance. In contrast, the importance of imperfections in the double layer model is minimized when analysis is performed in the frequency domain. Substantial variation in k(0) values was found by analysis of the 10 data sets for this highly surface-sensitive pathologically variable [Fe(CN)6](3-/4-) process, but remarkably, all fit the quasi-reversible model satisfactorily.

Quantitative Study of the Effect of Tissue Microstructure on Contraction in a Computational Model of Rat Left Ventricle

Tissue microstructure, in particular the alignment of myocytes (fibre direction) and their lateral organisation into sheets, is fundamental to cardiac function. We studied the effect of microstructure on contraction in a computational model of rat left ventricular electromechanics. Different fibre models, globally rule-based or locally optimised to DT-MRI data, were compared, in order to understand whether a subject-specific fibre model would enhance the predictive power of our model with respect to the global ones. We also studied the impact of sheets on ventricular deformation by comparing: (a) a transversely isotropic versus an orthotropic material law and (b) a linear model with a bimodal model of sheet transmural variation. We estimated ejection fraction, wall thickening and base-to-apex shortening and compared them with measures from cine-MRI. We also evaluated Lagrangian strains as local metrics of cardiac deformation. Our results show that the subject-specific fibre model provides little improvement in the metric predictions with respect to global fibre models while material orthotropy allows closer agreement with measures than transverse isotropy. Nonetheless, the impact of sheets in our model is smaller than that of fibres. We conclude that further investigation of the modelling of sheet dynamics is necessary to fully understand the impact of tissue structure on cardiac deformation.

A bidomain model of the ventricular specialized conduction system of the heart

An efficient bidomain model of electrical activity in cardiac specialized conduction system fibers is developed and applied to a geometric model of the specialized conduction system and the ventricles. The bidomain model allows the impact of externally applied electric fields on the specialized conduction system to be studied. To model this system, the three-dimensional bidomain equations for a fiber are reduced to one-dimensional equations governing electrical propagation by averaging over the fiber cross section. The one-dimensional equations are coupled to the surrounding three-dimensional extracellular electrical field to allow their use in defibrillation studies. A finite element method, with semi-implicit time stepping, is developed to numerically solve the equations. Current flow through fiber branch points is governed by Kirchhoffs law and imposed directly in a weak formulation of the equations for use with the finite element method. Coupling between intracellular potential in distal system Purki...

Development and Analysis of Patient-Based Complete Conducting Airways Models

The analysis of high-resolution computed tomography (CT) images of the lung is dependent on inter-subject differences in airway geometry. The application of computational models in understanding the significance of these differences has previously been shown to be a useful tool in biomedical research. Studies using image-based geometries alone are limited to the analysis of the central airways, down to generation 6–10, as other airways are not visible on high-resolution CT. However, airways distal to this, often termed the small airways, are known to play a crucial role in common airway diseases such as asthma and chronic obstructive pulmonary disease (COPD). Other studies have incorporated an algorithmic approach to extrapolate CT segmented airways in order to obtain a complete conducting airway tree down to the level of the acinus. These models have typically been used for mechanistic studies, but also have the potential to be used in a patient-specific setting. In the current study, an image analysis and modelling pipeline was developed and applied to a number of healthy (n = 11) and asthmatic (n = 24) CT patient scans to produce complete patient-based airway models to the acinar level (mean terminal generation 15.8 ± 0.47). The resulting models are analysed in terms of morphometric properties and seen to be consistent with previous work. A number of global clinical lung function measures are compared to resistance predictions in the models to assess their suitability for use in a patient-specific setting. We show a significant difference (p < 0.01) in airways resistance at all tested flow rates in complete airway trees built using CT data from severe asthmatics (GINA 3–5) versus healthy subjects. Further, model predictions of airways resistance at all flow rates are shown to correlate with patient forced expiratory volume in one second (FEV1) (Spearman ρ = −0.65, p < 0.001) and, at low flow rates (0.00017 L/s), FEV1 over forced vital capacity (FEV1/FVC) (ρ = −0.58, p < 0.001). We conclude that the pipeline and anatomical models can be used directly in mechanistic modelling studies and can form the basis for future patient-based modelling studies.

Integrated approach for the study of anatomical variability in the cardiac Purkinje system: From high resolution MRI to electrophysiology simulation

The ordered electrical stimulation of the ventricles is achieved by a specialized network of fibres known as the Purkinje system. The gross anatomy and basic functional role of the Purkinje system is well understood. However, very little is known about the detailed anatomy of the Purkinje system, its inter-individual variability and the implications of the variability in ventricular function, in part due to limitations in experimental techniques. In this study, we aim to provide new insight into the inter-individual variability of the free running Purkinje system anatomy and its impact on ventricular electrophysiological function. As a first step towards achieving this aim, high resolution magnetic resonance imaging (MRI) datasets of rat and the rabbit ventricles are obtained and analysed using a novel semi-automatic image processing algorithm for segmentation of the free-running Purkinje system. Segmented geometry from the MRI datasets is used to construct a computational model of the Purkinje system, which is incorporated in to an anatomically-based ventricular geometry to simulate ventricular electrophysiological activity.