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. 2015 Jan 6:5:511.
doi: 10.3389/fphys.2014.00511. eCollection 2014.

Cellular cardiac electrophysiology modeling with Chaste and CellML

Jonathan Cooper  1 , Raymond J Spiteri  2 , Gary R Mirams  1
Affiliations

Cellular cardiac electrophysiology modeling with Chaste and CellML

Jonathan Cooper et al. Front Physiol. .

Abstract

Chaste is an open-source C++ library for computational biology that has well-developed cardiac electrophysiology tissue simulation support. In this paper, we introduce the features available for performing cardiac electrophysiology action potential simulations using a wide range of models from the Physiome repository. The mathematics of the models are described in CellML, with units for all quantities. The primary idea is that the model is defined in one place (the CellML file), and all model code is auto-generated at compile or run time; it never has to be manually edited. We use ontological annotation to identify model variables describing certain biological quantities (membrane voltage, capacitance, etc.) to allow us to import any relevant CellML models into the Chaste framework in consistent units and to interact with them via consistent interfaces. This approach provides a great deal of flexibility for analysing different models of the same system. Chaste provides a wide choice of numerical methods for solving the ordinary differential equations that describe the models. Fixed-timestep explicit and implicit solvers are provided, as discussed in previous work. Here we introduce the Rush-Larsen and Generalized Rush-Larsen integration techniques, made available via symbolic manipulation of the model equations, which are automatically rearranged into the forms required by these approaches. We have also integrated the CVODE solvers, a 'gold standard' for stiff systems, and we have developed support for symbolic computation of the Jacobian matrix, yielding further increases in the performance and accuracy of CVODE. We discuss some of the technical details of this work and compare the performance of the available numerical methods. Finally, we discuss how this is generalized in our functional curation framework, which uses a domain-specific language for defining complex experiments as a basis for comparison of model behavior.

Keywords: C++; CellML; ODE; cardiac; electrophysiology; simulation; software.

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Figures

Figure 1
Figure 1
Single cell action potential traces for the Shannon et al. (2004) model using a range of numerical algorithms with different timesteps. Top: traces with eMRMS ⩽ 0.05 (overlapping), bottom: traces with eMRMS > 0.05. Note that nearly all the solvers in the upper figure diverged (failed to provide any solution) when running at larger timesteps or with more relaxed tolerances. It is the fact that the Generalized Rush–Larsen solvers are very stable that means that traces were produced even at large timesteps, so their dominance of the bottom plot should be seen as a demonstration of their stability rather than a statement on their accuracy.
Figure 2
Figure 2
Left: a comparison of the time taken to simulate one second of activity with CVODE and numerical Jacobians, for each model, under different compilers. The Intel/IntelProduction mostly coincide, as do the GccOpt/GccOptNative lines. The last point (Clancy and Rudy, 2002) is omitted for clarity but shows the same trend at wall times around 4.65 s. Right: the speed-up provided relative to gcc (debug). Models ranked according to the wall time required under the IntelProductionCvode build, as listed in Tables 1, 2.
Figure 3
Figure 3
Distributions of model simulation times for one second of electrophysiology for each different numerical method. The CVODE solvers (“CV–AJ” with analytic Jacobian, and “CV–NJ” with numerical approximation to the Jacobian) consistently outperform any other method by an order of magnitude or more.
Figure 4
Figure 4
Left: performance of CVODE with and without use of an analytic Jacobian. For seven models an analytic Jacobian was not stable, and these points are omitted (those shown in Tables 1, 2 as “—” in the CVODE AJ column). Right: relative speed when using analytic Jacobian rather than a numerical approximation; 10–20% speed-up is most common. Models ranked according to the wall time required for CVODE NJ under the IntelProductionCvode build, as listed in Tables 1, 2.
Figure 5
Figure 5
Speed of simulation when using lookup tables, relative to without lookup tables. Shown for all solvers with the “IntelProductionCvode” build. The black dashed line represents the same speed as without lookup tables; above is a speed-up; below is a slowdown. There are some gaps in the CVODE graphs to denote when analytic Jacobians are not available/stable (7 models), and also where the simulations with lookup tables failed to converge (2 additional models with analytic Jacobians, and 3 models with numerical Jacobians). Models ranked according to the wall time required for CVODE NJ under the IntelProductionCvode build, as listed in Tables 1, 2.
Figure 6
Figure 6
Total wall time (including PDE solution, output, etc.) for a simulated 500 ms of monodomain activity in a 1 cm strand of tissue with inter-node spacing of 0.01 cm, using the fastest compiler settings. Top: for a PDE timestep of 0.01 ms; bottom: for a PDE timestep of 0.1 ms. Missing data points indicate such small ODE timesteps were needed to reach a sufficiently converged solution that the simulation would have taken over 15 min (off the top of the scale).
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