Abstract
A mathematical method is presented that allows fast and simple computation of the electric field and current density induced inside a homogeneous spherical volume conductor by current flowing in a coil. The total electric field inside the sphere is computed entirely from a set of line integrals performed along the coil current path. Coils of any closed shape are easily accommodated by the method. The technique can be applied to magnetic brain stimulation and to magnetoencephalography. For magnetic brain stimulation, the total electric field anywhere inside the head can be easily computed for any coil shape and placement. The reciprocity theorem may be applied so that the electric field represents the lead field of a magnetometer. The finite coil area and gradiometer loop spacing can be precisely accounted for without any surface integration by using this method. The theory shows that the steady-state, radially oriented induced electric field is zero everywhere inside the sphere for ramping coil current and highly attenuated for sinusoidal coil current. This allows the model to be extended to concentric spheres which have different electrical properties.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramowitz, M. andStegun, I. A. (1964)Handbook of mathematical functions. Dover Publications, New York.
Amassian, V. E., Cracco, R. Q., andMaccabee, P. J. (1989) Focal stimulation of human cerebral cortex with the magnetic coil: a comparison with electrical stimulation.Electroenceph. Clin. Neurophysiol.,74, 401–416.
Arfken, G. (1985)Mathematical methods for physicists. Academic Press. Orlando, Florida.
Barker, A. T., Freeston, I. L., Jalinous, R. andJarratt, J. A. (1987) Magnetic stimulation of the human brain and peripheral nervous system: an introduction and the results of an initial clinical evaluation.Neurosurg.,20, 100–109.
Branston, N. M. andTofts, P. S. (1991) Analysis of the distribution of currents induced by a changing magnetic field in a volume conductor.Phys. in Med. & Biol.,36, 161–168.
Cohen, D., Cuffin, B. N., Yunokuchi, K., Maniewski, R., Purcell, C., Cosgrove, G. R., Ives, J., Kennedy, J. G. andSchomer, D. L. (1990) MEG versus EEG localization test using implanted sources in the human brain.Ann. Neurol.28, 811–817.
Cohen, D. andCuffin, B. N. (1991) Developing a more focal magnetic stimulator. Part I: Some basic principles.J. Clin. Neurophysiol.,8, 102–111.
Cohen, L. G., Roth, B. J., Nilsson, J., Dang, N., Panizza, M., Bandinelli, S., Friauf, W. andHallett, M. (1990) Effects of coil design on delivery of focal magnetic stimulation. Technical considerations.Electroenceph. Clin. Neurophysiol.,75, 350–357.
Cohen, L. G., Roth, B. J., Wassermann, E. M., Topka, H., Fuhr, P., Schultz, J. andHallett, M. (1991) Magnetic stimulation of the human cerebral cortex, an indicator of reorganization in motor pathways in certain pathological conditions.J. Clin. Neurophys.,8, 56–65.
Cuffin, B. N. andCohen, D. (1977) Magnetic fields of a dipole in special volume conductor shapes.IEEE Trans.,BME-24, 372–381.
Cuffin, B. N. andCohen, D. (1983) Effects of detector coil size and configuration on measurements of the magnetoencephalogram.J. Appl. Phys.,54, 3589–3594.
Ioannides, A. A. andSwithenby, S. J. (1987) An efficient magnetic flux integration method for bounded current sources.Ibid.,61, 4925–4927.
Jackson, J. D. (1975)Classical electrodynamics. John Wiley & Sons, New York.
Meyer, B. U., Britton, T. C., Kloten, H., Steinmetz, H. andBenecke, R. (1991) Coil placement in magnetic brain stimulation related to skull and brain anatomy.Electroenceph. Clin. Neurophysiol.,81, 38–46.
Plonsey, R. (1972) Capability and limitations of electrocardiography and magnetocardiography.IEEE Trans.,BME-19, 239–244.
Press, W. H., Flannery, B. P., Teukolsky, S. A. andVetterling, W. T. (1986)Numerical recipes. Cambridge University Press, New York.
Reilly, J. P. (1989) Peripheral nerve stimulation by induced electric currents: exposure to time-varying magnetic fields.Med. & Biol. Eng. & Comput.,27, 101–110.
Roth, B. J. andBasser, P. J. (1990) A model of the stimulation of a nerve fiber by electromagnetic induction.IEEE Trans.,BME-37, 588–597.
Roth, B. J., Saypol, J. M., Hallett, M. andCohen, L. G. (1991) A theoretical calculation of the electric field induced in the cortex during magnetic stimulation.Electroenceph. Clin. Neurophysiol.,81, 47–56.
Singh, S., Richards, W. F., Zinecker, J. R. andWilton, D. R. (1990) Accelerating the convergence of series representing the free space periodic Green's function.IEEE Trans.,AP-38, 1958–1962.
Stoy, R. D., Foster, K. R. andSchwan, H. P. (1982) Dielectric properties of mammalian tissues from 0.1 to 100 MHz: a summary of recent data.Phys. in Med. & Biol.,27, 501–513.
Tofts, P. S. (1990) The distribution of induced currents in magnetic stimulation of the nervous system.Ibid.,35, 1119–1128.
Ueno, S., Tashiro, T. andHarada, K. (1988) Localized stimulation of neural tissues in the brain by means of a paired configuration of time-varying magnetic fields.J. Appl. Phys.,64, 5862–5864.
Ueno, S., Matsuda, T. andFujiki, M. (1990) Functional mapping of the human cortex obtained by focal and vectorial magnetic stimulation of the brain.IEEE Trans.,MAG-26, 1539–1544.
Williamson, S. J. andKaufman, L. (1984) Frontiers in the new science of biomagnetism. InBiomagnetism: applications & theory.Weinberg, H., Stroink, G. andKatila, T. (Eds.), Pergamon Press, New York, 471–490.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eaton, H. Electric field induced in a spherical volume conductor from arbitrary coils: application to magnetic stimulation and MEG. Med. Biol. Eng. Comput. 30, 433–440 (1992). https://doi.org/10.1007/BF02446182
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02446182