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

Abstract

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.

Introduction

Computational modelling of cardiac electrophysiology has developed extensively over the last 50 years at all spatial scales. Numerous models at the cellular scale are now available, covering a range of species, cell types, and experimental conditions. As well as being used to study cellular scale phenomena, these models are frequently also embedded within a framework that describes cardiac tissue, in order to investigate the propagation of electrical activity that gives rise to the heartbeat. Such emergent behaviour at the tissue level can be very effectively modelled (Clayton et al., 2010).

With its long history, this field is relatively mature. Many of the cellular level models have been curated and are available in a standard computer-readable format from the CellML model repository (Lloyd et al., 2008), providing easy access to checked encodings of the model equations. Several tissue-level simulation environments are also available (Niederer et al., in press). However despite this, little has been written looking in detail at the interface between these levels. Perhaps this is because, as we shall describe shortly, this interface is relatively straightforward and well-defined from a mathematical point of view. However, in order to ensure correct operation of a cellular level model within a tissue simulation, there are many details and potential inconsistencies that must be taken into account and addressed.

Tissue-level cardiac electrophysiology is usually modelled using the bidomain equations (or the monodomain simplification thereof). These consist of two partial differential equations (PDEs) describing the intracellular and extracellular potential fields (ϕ_i and ϕ_e) through the cardiac tissue as a reaction–diffusion system, coupled at each point in space with a system of ordinary differential equations (ODEs) representing the concentrations of ions and other variables at the cellular level (see, for example, Keener and Sneyd, 1998). Let Ω denote the region occupied by the cardiac tissue. In the parabolic–elliptic formulation, which describes ϕ_e and the transmembrane voltage V_m = ϕ_i − ϕ_e, the bidomain equations are then

$$\chi \left(C_m\frac{\partial V_m}{\partial t}+I_{\mathrm{ion}}\left(\mathit{u},V_m\right)\right)-\nabla ·\left({\sigma }_i\nabla \left(V_m+{\varphi }_e\right)\right)=I_i^{\left(\mathrm{vol}\right)},$$

$$\nabla ·\left(\left({\sigma }_i+{\sigma }_e\right)\nabla {\varphi }_e+{\sigma }_i\nabla V_m\right)=-I_{\mathrm{total}}^{\left(\mathrm{vol}\right)},$$

$$\text{and}\text{for}\text{each}\text{point}\text{in}\Omega ,\frac{\partial \mathit{u}}{\partial t}=\mathit{f}\left(\mathit{u},V_m\right),$$

where (1a) and (1b) describe the spatial and temporal evolution of the electrical potentials, and (1c) describes the remaining cellular level behaviour. Here σ_i is the intracellular conductivity tensor, σ_e is the extracellular conductivity tensor, χ is the surface area to volume ratio, $C_m$ is the membrane capacitance per unit area, u is a set of cell-level variables (such as gating variables, ion concentrations, etc.), and I_ion ≡ I_ion(u, V_m) is the transmembrane ionic current per surface area. $I_{\mathrm{total}}^{\left(\mathrm{vol}\right)}=I_i^{\left(\mathrm{vol}\right)}+I_e^{\left(\mathrm{vol}\right)}$, where the source terms $I_i^{\left(\text{vol}\right)}$ and $I_e^{\left(\text{vol}\right)}$ are the intra- and extracellular stimuli per unit volume. Appropriate boundary and initial conditions must also be applied; see (Pathmanathan et al., 2010a) for details.

In this paper, we examine three classes of issues concerning the interoperability of CellML models of cellular electrophysiology within cardiac tissue simulations, and indicate possible strategies to address each of them.

The first class of issues arises from the fact that the cellular models available in the CellML model repository are not formulated as components of a tissue model, but represent entities somewhere between an isolated cell and a patch of cell membrane. They thus do not provide straightforward definitions of the terms I_ion and f as they appear in (1). Rather, their equations appear in the form

$$\frac{d{V}_m}{dt}=-\frac{I_{\text{ion}}\left(\mathit{u},V_m\right)+I_{\text{stim}}}{C_m},$$

$$\frac{d\mathit{u}}{dt}=\mathit{f}\left(\mathit{u},V_m\right),$$

or a variation thereof. The first challenge is therefore to identify the relevant variables within the cell model for coupling to the tissue model, and extract the necessary equations. This is the topic of Section 2.1.

The next class of issues, treated in Section 2.2, is that of inconsistencies between the models being connected. This may arise through differing use of units, from variations in how the models are structured, or from differences in parameter values. The final class of issues we address, in Section 2.3, are those faced by software that is intended to be generic enough to be able to deal with different cell types. A sample simulation, illustrating the importance of addressing these issues correctly, is presented in Section 3.

These issues are of particular importance to those seeking to exploit the full potential of having cellular models available in a standard language such as CellML, by being able to reuse these models within tissue simulations without significant manual effort. For each issue, we therefore also describe the support available in Chaste (Pitt-Francis et al., 2009), via the PyCml software (Cooper, 2009, Garny et al., 2008), for automatically addressing it when processing a suitably annotated cell model. This allows the transparent inclusion of any cell model represented in CellML within a tissue simulation. The cardiac portion of Chaste is a powerful, efficient, parallel and well-engineered monodomain/bidomain solver. Chaste is open-source and available for download at http://web.comlab.ox.ac.uk/chaste/. All of the functionality described in this paper is present in the version 2.2 release.

We conclude in Section 4 by identifying common threads arising from these issues, and discuss some of the wider implications for the computational modelling field.