Single Cell Scale

For electrophysiological simulation, the basic unit of the model is the cellular activity. Cellular models of cardiac electrical activity were first constructed over 50 years ago, and today, these models have been refined and are routinely put into organ scale simulations. The modelling components are described below.

Ionic models refer to the electrical representation of the cell based on the movement of ions across the cell membrane, resulting in a change in the voltage across the membrane. Schematically, a capacitor is placed in parallel with a number of ionic transport mechanisms. In general, an ionic model is of the form

\( \begin{equation} I_{\rm m} = C_m \frac{d V_{\rm m}}{d t} + \Sigma_\chi I_\chi \nonumber \end{equation} \)

where \(V_{\rm m}\) is the voltage across the cell membrane, \(I_{\rm m}\) is the net current across the membrane, \(C_{\rm m}\) is the membrane capacitance, and \(I_{\chi}\) is a particular transport mechanism, either a channel, pump or exchanger. Additional equations may track ionic concentrations and the processes which affect them, like calcium-induced release from the sarcoplasmic reticulum. By convention, outward current, i.e., positive ions leaving the cell, is defined as being positive, while inward currents are negative.

Channels are represented as resistors in series with a battery. The battery represents the Nernst Potential resulting from the electrical field developed by the ion concentration difference across the membrane. It is given by

\( \begin{equation} E_S = \frac{RT}{zF}\ln\frac{[S]_i}{[S]_e} \nonumber \end{equation} \)

where \(R\) is the gas constant, \(T\) is the temperature, \(F\) is Faraday's constant and \(z\) is the valence of the ion species \(S\).

Channels are dynamical systems which open and close in response to various conditions like transmembrane voltage, stretch, ligands, and other factors. The opening and closing rates of the channels vary by several orders of magnitudes, and are generally, nonlinear in nature. Thus, the equivalent electrical resistance of a channel is a time dependent quantity.

The first representation of a cardiac cell was that produced by D. Noble of a Purkinje cell, based on modification of the Hodgkin-Huxley nerve action potential. Since then, hundreds of models have been developed for many reasons. Ionic models need to be developed to match experimental procedures if the models are to be predictive and offer insight into mechanistic workings. In mammals, action potential durations range from tens of milliseconds for mice to several hundreds of milliseconds for large animals. Atrial myocyte protein expression is quite different from that of ventricular myocytes, resulting in different action potential durations and shapes. Even nearby cells exhibit action potential differences due to slightly different levels of channel protein expression. The complexity of ionic models has been steadily growing as more knowledge is gained through better experimental techniques, equipment and specific blockers of transport mechanisms. As more mechanisms are identified as affecting electrophysiology, either directly or indirectly, they are incorporated into ionic models. Selection of the model to use is not always obvious as different models may be available for the same species and heart location, which may yield quite different behaviour. Regardless, identifying the strengths and weaknesses of an ionic model for a particular application is important.

Ionic models may be biophysically detailed or phenomenological. Biophysically detailed models attempt to discretely depict important currents and processes within the cell. Early models had approximately ten equations depicting only sodium and potassium channels, while present models have hundreds of equations taking into account not only membrane ion transporters, but intracellular calcium handling, mitochondrial function, and biochemical signalling pathways. Conversely, phenomenological ionic models use a set of currents which faithfully reproduce whole cell behaviour in terms of action potential shape and duration, and restitution properties. The currents in the phenomenological model are not physiological but can be considered as amalgamations of known currents. While these models are computationally much simpler and easier to tune, they lose the direct correspondence with the biophysics which makes implementing drug effects or cellular pathologies challenging. Finally, cellular automata models have also been used, and can be considered as a type of phenomenological model. These models do not use differential equations but have a set of rules which dictate transitions between discrete states of the model. As such, these models are computationally light, and can be as detailed as required. However, behaviour may not be as rich as differential equations. The choice of model, biophysical or phenomenological, depends on the nature of the problem being considered, and the availability of computational resources for the problem size.

Ionic models A: Myocyte schematic showing the membrane (blue) which contains channels (yellow cylinders), pumps (yellow circles) and exchangers (blue circles). The sarcoplasmic reticulum is divided into the junctional (JSR) and network (NSR) regions and channels and pumps control ion movement into and out of it. Figure taken from Michailova et al.. B: Electrical representation of a cell showing a limited number of currents: fast sodium (\(I_{\rm {Na}}\)), inward rectifier (\(I_{\rm {K1}}\)), fast and slow repolarizing currents (\(I_{\mathrm{Kr}}\) and \(I_{\mathrm{Ks}}\)) L-type calcium (\(I_{\mathrm{Ca,L}}\)), the sodium-potassium pump (\(I_{\mathrm{NaK}}\)) and the sodium-calcium exchanger (\(I_{\mathrm{NCX}}\)). Nernst potentials for the ionic species are indicated (\(E_{\rm S}\)) as is the membrane capacitance (\(C_{\rm m}\)). C: Effect of applying a suprathreshold current at time 0 to the tenTuscher ionic model of the human ventricular myocyte. Top: Major ionic currents. Note that the following currents are clipped: \(I_{\mathrm {Na}}\) (-46.4), \(I_{\mathrm{to}}\) (15.3), and \(I_{\mathrm{Ca,L}}\) (-9.6). Bottom: the resulting action potential (\(V_{\rm m}\)) and myoplasmic calcium transient (\([Ca_{\rm i}]\)).

When considering the mathematical description of ionic channels, there are two main approaches which are detailed below.