Modelling Electrical Stimulation of Tissue

Abstract

This chapter describes the theory and techniques for modelling the electrical activity of excitable cells and tissues, along with their electrical simulation, using COMSOL. It begins with a summary of Maxwell’s equations, before moving on to electrostatics and volume conductor theory. Examples in COMSOL are presented for designing an optimal electric field stimulator for cell cultures in a Petri dish, as well as determining the current density and access resistance of an isopotential disc electrode in an infinite medium. The chapter then proceeds to cover continuum models of excitable tissues such as nerve and muscle and their electrical stimulation, expressed as classical bidomain and monodomain formulations. Examples in COMSOL are presented for modelling reentrant spiral waves in a slab of cardiac tissue and a propagating action potential in a nerve axon embedded in a nerve bundle stimulated by extracellular cuff electrodes.

Problems

6.1

Two platinum spherical stimulating electrodes of radius 0.5 mm are placed 2 mm apart centre-centre in an infinite saline solution of conductivity 1 S m$^{-1}$. Assuming the surface of each electrode lies at an isopotential state, use COMSOL to determine the resistance between the electrodes.

6.2

Consider a 2D square slab of excised cardiac tissue of sidelength 1 cm, as shown below. Electrical stimulating electrodes are placed on its left and right boundaries such that the left boundary is held at a potential of 1 V whilst the right is at ground. All other boundaries are assumed to be electrically insulated. The tissue consists of parallel muscle fibres oriented at an angle $\theta$ relative to the x-axis, with longitudinal conductivity (i.e. along the fibre) of 0.2 mS cm$^{-1}$ and transverse conductivity 0.1 mS cm$^{-1}$. Plot the potential distribution and current streamlines for fibre angles $\theta = 0^{\circ }$, $\theta = 45^{\circ }$ and $\theta = 90^{\circ }$.

6.3

A platinum disc electrode of radius 1 mm is injecting current into a hemispherical infinite saline domain of conductivity 1 S m$^{-1}$, similar to the geometry shown in Fig. 6.5. At the junction between the saline and the platinum, there is a liquid-metal interface, represented by a distributed resistance of $0.001\,\Omega \,\mathrm {m}^2$. If the platinum electrode is at a steady-state equipotential level of 1 V, the local potential in the saline adjacent to the electrode will be less, owing to the voltage drop across the distributed resistance. Using COMSOL, plot the inward current density as a function of radial position along the disc, comparing against the theoretical solution with no distributed resistance .