Volume conductors

FIGURE 19.1 Bioelectric current loop, concept drawing. Current enters the membrane atz = zi, flows intracellularly and emerges at Z2. Flow in the extracellular volume may be along the membrane (solid) or throughout the surrounding volume conductor (dashed). 

In most living systems, however, extracellular currents extend throughout a more extensive volume conductor, as suggested by the dashed lines for 4 in Figure 19.1. Their direction and magnitude must be found as a solution to Poisson s equation in the extracellular medium, with the sources and sinks around the active medium used as boundary conditions. In this context, Poisson s equation becomes (Plonsey and Barr, 2000]... 

Note the duality Equation 19.6 and the form of Poisson s equation commonly used in problems in electrostatics. There, V /(—p/e), where p is the charge density, and s is the permittivity. Recognition of this duality is useful not only in locating solution methods for bioelectric problems, since the mathematics is the same, but also in avoiding confusion between electrostatics and problems of extracellular current flow through a volume conductor. The problems obviously are physically quite different (e.g., permittivity s is not conductivity ere)-... 

Extracellular waveforms are created by the flow of current through the volume conductor surrounding the active membrane. An expression for the extracellular potential 0e, outside a cylindrical fiber with a large conducting volume was given by Spach et al. [1973] as... 

Extracellular potentials Potentials generated between two sites outside the membrane (external to active cells), for example, between two sites on the skin, due to current flow in the volume conductor. 

Volume conductor The electrically conductive interior region of the body surrounding electrically active membrane. 

Basic Relations in the Idealized Homogeneous Volume Conductor... 

Excitable tissue, when activated, will be found to generate currents both within itself and also in aU surrounding conducting media. The latter passive region is characterized as a volume conductor. The adjective volume emphasizes that current flow is three-dimensional, in contrast to the confined onedimensional flow within insulated wires. The volume conductor is usually assumed to be a monodomain (whose meaning will be ampHfied later), isotropic, resistive, and (frequently) homogeneous. These are simply assumptions, as will be discussed subsequently. The permeabihty of biologic tissues is important... 

If one imagines an infinitely thin wire insulated over its extent except at its tip to be introducing a current into a uniform volume conductor of infinite extent, then we illustrate an idealized point source. Assuming the total applied current to be To and located at the coordinate origin, then by symmetry the current density at a radius r must be given by the total current To divided by the area of the spherical surface, or... 

Effects of Volume Conductor Inhomogeneities Secondary Sources and Images... 

In regions of tissue where there are no sources, S is zero. In these cases, the divergenceless of / is equivalent to the law of conservation of current that is often invoked when analyzing electrical circuits. Another property of a volume conductor is that the current density and the electric field, E (V/m), are related linearly by Ohm s Law,... 

Before discussing the conductivity of tissue, consider one of the simplest and most easily understood volume conductors sahne. The electrical conductivity of sahne arises from the motion of free ions in response to a steady electric field, and is on the order of 1 S/m. Besides conductivity, another property of saline is its electrical permittivity, s (S sec/m). This property is related to the dielectric constant, /c (dimensionless), by e = /cso, where Sq is the permittivity of free space, 8.854 x 10 S sec/m. Dielectric properties arise from bound charge that is displaced by the electric field, creating a dipole. They can also arise if the appHed electric field aligns molecular dipoles (such as the dipole moments of water molecules) that are normally oriented randomly. The DC dielectric constant of sahne is similar to that of water (about... 

Conservation of current A fundamental law of electrostatics, stating that there is no net current entering or leaving at any point in a volume conductor. 

Volume conductor A three-dimensional region of space containing a material that passively conducts electrical current. 

We begin by stating the mathematical formulation for a bioelectric volume conductor, continue by describing the model construction process, and foUow with sections on numerical solutions and computational considerations. We conclude with a section on error analysis coupled with a brief introduction to adaptive methods. 

As noted in the chapter on Volume Conductor Theory, most bioelectric field problems can be formulated in terms of either the Poisson or the Laplace equation for electrical conduction. Since Laplace s equation is the homogeneous counterpart of the Poisson equation, we will develop the treatment for a general three-dimensional Poisson problem and discuss simplifications and special cases when necessary. 

A typical bioelectric volume conductor can be posed as the following boundary value problem ... 

For volume conductor problems, A contains aU of the geometry and conductivity information of the model. The matrix A is symmetric and positive definite thus, it is nonsingular and has a unique solution. Because the basis function differs from zero for only a few intervals, A is sparse (only a few of its entries are nonzero). 

We now illustrate the concepts of the FE method by considering the solution of Equation 23.1 using linear 3D elements. We start with a 3D domain 2 which represents the geometry of our volume conductor and break it up into discrete elements to form a finite dimensional subspace, 2 ,. For 3D domains we have the choice of representing our function as either tetrahedra. 

The example network in Figure 23.1, is solving a bioelectric field problem for a dipolar source in a volume conductor model of a head. The domain is discretized with linear tetrahedral finite elements, with five different conductivity types assigned through the volume. The problem is numerically approximated with a linear system, and is solved using the CG method. A set of virtual electrode points are rendered as pseudocolored spheres, to visualize the potentials at those locations on the scalp, and an iso-potential surface and several pseudocolored electric field streamlines are also shown. 

FIGURE 25.4 Simulated currents and extracellular potentials of frog sartorius muscle fiber (radius a = 50 /xm). (a) The net fiber current density is the summation of the current density through the sarcolemma and that passing the tubular mouth, (b) Extracellular action potentials calculated at increasing radial distances (in units of fiber radius) using a bidomain volume conductor model and the net current source in panel (a). The time axes have been expanded and truncated. 

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