The transient subthreshold response of spherical and cylindricalcell models to extracellular stimulation

The effect of extracellular stimulation on excitable tissue is evaluated using analytical models. Primary emphasis is placed on the determination of the rate of rise of the membrane potential in response to subthreshold stimulation. Three models are studied: 1) a spherical cell in a uniform electric field, 2) an infinite cylindrical fiber with a point source stimulus, and 3) a finite length cable with sealed ends and a stimulus electrode at each end. Results show that the rate of rise of the transmembrane potential was more rapid than the step response of a space-clamped membrane for all geometries considered. The response of the cylindrical fiber to extracellular stimulation is compared to previously reported studies of the cylindrical fiber response to intracellular stimulation. It is found that the location of the stimulus has little effect on the infinite fiber response. For terminated cables, however, an accurate model of stimulus response must discriminate between intracellular and extracellular stimulation.     

Membrane current from transmembrane potentials in complex core-conductor models

Core-conductor models, used to integrate the behavior of the longitudinal currents with the distributed voltages of electrically active tissue, have evolved for over a century. A critical step in the use of such models is the computation of membrane current from the set of distributed transmembrane potential values that exist at a given moment, where the potentials are obtained either experimentally or computationally. Over time, interest has developed in a number of substantial extensions of the original model to include such features as nonuniform spatial resistances, loop instead of linear structure, and multiple sites of extracellular stimulation. This paper concisely restates and extends the equations for calculation of transmembrane currents with the systematic inclusion of alternative cases, noting how they reduce to the standard forms. An important issue is how complex the calculation of membrane current has to be. Thus, the paper goes on to show criteria (based on the uniformity of resistance and the presence of stimulation) for deciding when membrane currents can be obtained with a relatively simple calculation with a single equation involving local variables versus with a more complex calculation involving the simultaneous solution of a (possibly large) set of equations.     

Status and trends in biomedical engineering education

An overview of graduate and undergraduate programs is given, including a brief history of each. With respect to undergraduate studies, the issues of accreditation, department status versus option programs, employment, registration, and medical and professional school opportunities are examined. Enrollment trends are examined. A survey of biomedical engineering textbooks and a bibliography containing 102 references are given in an appendix.     

Electric field stimulation of excitable tissue

Examines the transmembrane voltage response of an unmyelinated fiber to a stimulating electric field from a point current source. For subthreshold conditions, analytic expressions for the transmembrane potential, v_m, are developed that include the specific effects of fiber-source distance, h, and time from the onset of the stimulus, T. Suprathreshold effects are determined for two examples by extending the analytical results with a numerical model. The v_m , response is a complex evolution in time, especially for small h, that differs markedly from the “activating function”. In general, the subthreshold response is a good predictor of the wave shape of the suprathreshold v_m but a poor predictor of its magnitude. The subthreshold response also is a good (but not a precise) predictor of the region where excitation begins     

Theoretical Consideration for a Multipole Probe in Electrocardiographic Studies

This paper describes the design of an active source capable of approximating the three independent dipole and five independent quadrupole source components. The entire system utilizes a physical arrangement of 11 ``point sources' located at the intersections of a cubical lattice. By an appropriate selection of point sources and sinks and by proper control of their relative strengths, each of the eight aforementioned source components can be excited.     

The Eccentric Spheres Model as the Basis for a Study of the Role of Geometry and Inhomogeneities in Electrocardiography

To study the role of internal geometry and inhomogeneities in the "forward problem" of electrocardiography (ECG), a mathematical model was constructed which permitted manipulation of these variables. The model, which consists of two eccentric systems of concentric spheres, contains all the important torso compartments, namely the blood cavity, the myocardium, the pericardium, the lung region, the surface muscle layer, and the subcutaneous fat. An analytic solution is found in the form of a double series expansion in Legendre polynomials. The integrated effect of the inhomogeneities on the surface potential distribution is investigated. The model demonstrates the importance of interactions between the various torso components in determining the potential distribution at the surface.     

Capability and Limitations of Electrocardiography and Magnetocardiography

The capabilities and limitations of electrocardiography and magnetocardiography are discussed. Representing the electrical activity of the heart by an impressed current density ji, electrocardiography determines the spherical harmonic multipole expansion of its divergence (flux source), while magnetocardiography determines the spherical harmonic multipole expansion of the radial component of its curl (vortex source).