Calculation of the magnetic field due to a bioelectric current dipole in an ellipsoid
Tóm tắt
The bioelectric current dipole model is important both theoretically and computationally in the study of electrical activity in the brain and stomach due to the resemblance of the shape of these two organs to an ellipsoid. To calculate the magnetic field B due to a dipole in an ellipsoid, one must evaluate truncated series expansions involving ellipsoidal harmonics
$$
\mathbb{E}_n^m
$$
, which are products of Lamé functions. In this article, we extend a strictly analytic model (G. Dassios and F. Kariotou, J. Math. Phys. 44 (2003), 220–241), where B was computed from an ellipsoidal harmonic expansion of order 2. The present derivations show how the field can be evaluated to arbitrary order using numerical procedures for evaluating the roots of Lamé polynomials of degree 5 or higher. This can be accomplished using an optimization technique for solving nonlinear systems of equations, which allows one to acquire an understanding of the truncation error associated with the harmonic series expansion used for the calculation.
Tài liệu tham khảo
S. Baillet, J. C. Mosher, R. M. Leahy: Electromagnetic brain mapping. IEEE Signal Proc. Mag. 18 (2001), 14–30.
W. E. Byerly: An Elementary Treatise on Fourier Series and Spherical, Cylindrical and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. Ginn, Boston, 1893.
M. Camilleri, W. L. Hasler, H. P. Parkman, E. M. M. Quigley, E. Soffer: Measurement of gastrointestinal motility in the GI laboratory. Gastroenterol 115 (1998), 747–762.
D. Cohen: Magnetoencephalography—evidence of magnetic fields produced by alpha-rhytmic currents. Science 161 (1968), 784–786.
D. Cohen, E. A. Edelsack, J. E. Zimmerman: Magnetocardiograms taken inside a shielded room with a superconducting point-contact magnetometer. Appl. Phys. Lett. 16 (1970), 278–280.
D. Cohen: Magnetoencephalography—detection of brain’s electrical activity using a superconducting magnetometer. Science 175 (1972), 664–666.
A. Corrias, M. L. Buist: A quantitative model of gastric smooth muscle cellular activation. Physiol. Meas. 25 (2007), 849–861.
G. Dassios, F. Kariotou: Magnetoencephalography in ellipsoidal geometry. J. Math. Phys. 44 (2003), 220–241.
B. O. Familoni, T. L. Abell, K. L. Bowes: Model of gastric electrical activity in health and disease. IEEE Trans. Biomed. Eng. 42 (1995), 647–657.
W. E. Featherstone, M. C. Dentith: A geodetic approach to gravity data reduction for geophysics. Comput. Geosci. 23 (1997), 1063–1070.
M. S. Hämäläinen, J. Sarvas: Feasibility of the homogeneous head model in the interpretation of neuromagnetic fields. Phys. Med. Biol. 32 (1987), 91–97.
M. S. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila, O. Lounasmaa: Magnetoencephalography—theory, instrumentation and applications to noninvasive studies of the working human brain. Rev. Mod. Phys. 65 (1993), 413–497.
E. W. Hobson: The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, Cambridge, 1931.
A. Irimia, L. A. Bradshaw: Theoretical ellipsoidal model of gastric electrical control activity propagation. Phys. Rev. E 68 (2003).
A. Irimia: Electric field and potential calculation for a bioelectric current dipole in an ellipsoid. J. Phys. A, Math. Gen. 38 (2005), 8123–8138.
A. Irimia: Ellipsiodal electrogastrographic forward modelling. Phys. Med. Biol. 50 (2005), 4429–4444.
F. Kariotou: Electroencephalography in ellipsoidal geometry. J. Math. Anal. Appl. 290 (2004), 324–342.
J. Malmivuo, R. Plonsey: Bioelectromagnetism. Oxford University Press, New York, 1995.
M. P. Mintchev, K. L. Bowes: Conoidal dipole model of electrical-field produced by the human stomach. Med. Biol. Eng. Comput. 33 (1995), 179–184.
M. P. Mintchev, S. J. Otto, K. L. Bowes: Electrogastrography can recognize gastric electrical uncoupling in dogs. Gastroenterol 112 (1997), 2006–2011.
N. Mirizzi, R. Stella, U. Scafoglieri: A model of extracellular waveshape of the gastric electrical activity. Med. Biol. Eng. Comput. 23 (1985), 33–37.
N. Mirizzi, R. Stella, U. Scafoglieri: Model to simulate the gastric electrical control and response activity on the stomach wall and on the abdominal surface. Med. Biol. Eng. Comput. 24 (1986), 157–163.
Y. C. Okada, M. Lauritzen, C. Nicholson: MEG source models and physiology. Phys. Med. Biol. 32 (1987), 43–51.
J. W. Phillips, R. M. Leahy, J. C. Mosher, B. Timsari: Imaging neural activity using MEG and EEG. IEEE Eng. Med. Biol. Mag. 16 (1997), 34–42.
S. Ritter: The nullfield method for the ellipsoidal Stokes problem. J. Geod. 72 (1998), 101–106.
G. Romain, J. P. Barriot: Ellipsoidal harmonic expansions of the gravitational potential: theory and applications. Celest. Mech. Dyn. Astr. 79 (2001), 235–275.
E. B. Saff, A. D. Snider: Fundamentals of Complex Analysis with Applications to Engineering and Science, third edition. Prentice Hall, Upper Saddle River, 2003.
J. Sarvas: Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys. Med. Biol. 32 (1987), 11–22.
A. Sommerfeld: Electrodynamics. Academic Press, New York, 1952.