418
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38. NO. 5. MAY I Y Y I
Prediction of Magnetically Induced Electric Fields in Biological Tissue Kent R. Davey, Senior Member, IEEE, Chin Huei Cheng, and Charles M. Epstein
Abstract-There are many potential medical applications in which it is desirable to noninvasively induce electric fields. One such application that serves as the backdrop of this work is that of stimulating neurons in the brain. The magnetic fields necessary must be quite high in magnitude, and fluctuate rapidly in time to induce the internal electric fields necessary for stimulation. Attention is focused on the calculation of the induced electric fields commensurate with rapidly changing magnetic fields in biological tissue. The problem is not a true eddy current problem in that the magnetic fields induced do not influence the source fields. Two techniques are introduced for numerically predicting the fields, each employing a different gauge for the potentials used to represent the electric field. The first method employs a current vector potential lanal_ogous to A in classical magnetic field theory where V X A = B ) and is best suited to two-dimensional (2-D) models. The second represents the electric field as the sum of a vector plus the gradient of a scalar field; because the vector can be determined quickly using Biot Savart (which for circular coils degenerates to an efficient evaluation employing elliptic integrals), the numerical model is a scalar problem even in the most complicated three dimensional geometry. These two models are solved for the case of a circular current carrying coil near a conducting body with sharp corners.
I. INTRODUCTION/BACKGROUND T PRESENT, extracranial magnetic stimulation is used to study human motor systems [1]-141; preliminary research is directed towards visual and cognitive physiology [5]. Typical magnetic stimulators produce peak currents of several thousand amperes in brief pulses lasting 50-200 ps. Such currents are produced by capacitor discharge at 700-4000 V [6]-[8]. The energy per stimulus is high, and the efficiency of coupling to the brain is low The locus of cerebral motor stimulation appears to lie near the junction of white and gray matter, at least 4-5 mm below the cortical surface and 17 mm beneath the scalp [9], [lo]. There has been no simple technique for predicting the induced electric fields. Without detailed analytical calculations of the field, it is difficult to charac-
A
Manuscript received November 14, 1989; revised June 1 1 , 1990. This work was supported in part by a grant from the Emory-Georgia Tech Biomedical Research Foundation, in part by the Georgia Institute of Technology, Emory University, School of Medicine, and in part by the Rehabilitation Research and Development Center of the Atlanta VAMC. K . R. Davey and C . H . Cheng are with the School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332. C. M . Epstein is with the Department of Neurology, School of Medicine, Emory University, Atlanta, GA 30322. IEEE Log Number 9144692.
terize the strength and orientation of the neuronal stimulus, or to optimize the design of the stimulus coils. The ultimate objective is to produce a magnetic stimulator that can activate the cortex at high frequency, and thus facilitate applications in cognitive physiology [ 1 11. The fields so generated must be accurately quantified by position. Presented here is an alternative analysis approach to the conventional finite element procedure {hat seeks a threedimensiota1 (3-D) forqulation of the H field or potential 9 where H = -V(+); E is then typically realized through the curl of the magnetic field. Thus, the principal contribution of this work is the presentation of a fast, accurate, and efficient technique for solving the internal electric field induced by a rapidly changing magnetic field. In this derivation we assume a homogeneous, mildly conducting medium, ignoring the presence of the skull. This simplification is justified because the conductivity of the bone is much lower than that of the brain; the side of stimulation appears to lie beneath the brain surface [9], [lo], and there is no physiological reason to believe that the local currents which stimulate neurons extend appreciably outside the cortex. Magnetic field stimulation of the brain (soft tissue) is characterized by the following: characteristic length I = 0.1 m permeability y = 4.rr . IO-’ H/m dielectric constant E = 10‘ = 1/36.rr . 10’ F/m conductivity U = 0.06 S/m characteristic system time 7 = l/frequency = 0.1 ms electrical relaxation time 7, = € / U = 160 ys magnetic diffusion time 7, * = 0.7 ns = 300 ns electromagnetic time 7,,,,= quasistatic parameter p = {7,,,/7}’ = 1.0 x
x;
As shown by Melcher [13], a system with the aforementioned characteristic times is magnetoquasi-static and is represented by the following equations:
v
x H =
V
X
%E 7
+
E
=
+p
g]
ai
--.
at
at the frequencies characterizing The magnitude of magnetic field stimulation is sufficiently small that the second term on the right of (1) can be ignored without any
0018-9294/91/0500-0418$01.00 0 1991 IEEE
DAVEY et a!.: MAGNETICALLY INDUCED ELECTRIC FIELDS IN BIOLOGlCAL TISSUE
loss of accuracy. The value of will be even smaller at higher frequencies due to the decrease of the dielectric constant [ 141, [ 151. By taking the divergence of (1) and recalling the fact that the divergence of the curl of a vector is zero, the equation of interest in the tissue reduces to
v
*
OB =
v . 7 = 0.
11. ELECTRIC FIELDPREDICTION We will discuss two methods for predicting the induced electric fields from the time changing B fields. The first employs a current vector potential and is especially suited to two dimensional simulations. In three dimensions this technique would necessitate a vector field solution with a large number of unknowns. The second technique represents the electric field as another vector plus the gradient of a scalar. The vector part is found and expressed immediately in terms of the magnetic vector potential of the source coil. The scalar satisfies Laplace’s equation and is determined to insure that the boundary conditions on the current density are realized. This method is suitable for 2-D and 3-D problems, and always involves scalars only.
A. Current Vector Potential Approach We begin this procedure by representing the current density as the curl of a vector
7-vxF
1 cm
/
1 in dia Coil
(3)
Calculations of induced current densities have been approached a number of ways to date. Chiba er al. [16] attack the problem via a finite-element analysis involving the minimization of an energy functional. Because they are concerned about 60 Hz transmission line electric field effects, they do not use magnetoquasistatics; their primary focus is the external electric field of the line. Like the work by Den0 and Kaune [17], [18], they examine surface charge effects and the internal currents accompanying changes of these surface charges. Considerable attention must be given to exposed tissue surface areas, an emphasis that is peripheral to the problem as viewed here in its eddy current context. Spiegel [19], [20] attempt a solution of the complete Maxwell Equations (even at 60 Hz) in terms of the incident and scattered field.
(4)
With this assignment we are guaranteed a divergence free current density. In terms of the resistivity p = l / a , (2) can be written as
The magnetic field density B is caused solely by the source currents in the excitation coil. Fig. 1 shows a test geometry for the induced field prediction. In practice the coil is frequently placed flat against the heat; this geometry serves to better delineate the differences between the two techniques under discussion in this paper. Note that we have deliberately chosen a square rather
-+ I 419
Conducting Medium conductivity 1 Slm
Source Current 10 t u r n s ramp up in 5 0 p s
-
4 in
/ In
Fig. 1 . Test geometry for the determination of the induced electric field generated by a coil ramping up to 31 536 amp turns in 50 ps. The conductivity of the tissue is taken here to be I S/m. The square cross-section conducting medium is assumed to be long in extent with the same crosssection for all z . The excitation coil is a simple current loop existing only a t z = 0.
than circular geometry as an exceptionally rigorous test of the techniques. The square corners contort the current paths, and thus tax the numerical model more than a cylindrical or spherical model would. In a homogeneous conducting region, (5) reduces to
-
V2F =
ai
0-.
at
This equation may be discretized using either a finite difference or finite-element procedure. We employed a finite difference impLementation of (6) and enforce the condition that ri llJll = 0, i.e., that no normal current exists on the tissue air interface. I ) Source Coil Field Prediction-Biot-Savart: The principal issue is how to calculate the main source field that drives ( 5 ) . The generalized Biot-Savart rule involves the integration of the source current:
(7) Although this formulation always works, it can be time consuming. Recall that for either a finite difference or finite-element implementation, the source term must be evaluated at every node point in the grid. In three dimensions the overhead becomes intolerable. 2) Source Coil Field Predictions-Elliptic Integrals: The fields for circular coils are more easily solved using elliptic integrals. Smythe [21] shows that for a coil of radius a carrying I amps, the magnetic field density at some radial displacement p and axial position z above the coil is
. j-K(k) +
+
a2 p2 + z 2 (a - p)2 + z 2
420
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38. NO. 5. MAY 1991
(9) where k2
=
m=
4a
P
(a
+ p)? + z 2 .
1 - J 1 - k2 1
+ m’
Abramowitz and Stegun [221 provide an nition of elliptic integrals that are easily evaluated in terms of polynomial approximations:
Ko(m) = (ao
+ a l m l + u2m3
=
1.3862944
bo
=
0.5
U! =
0.11 19723
61
=
0.1213478
62 = 0.0288729
(10)
+ alml + a 2 m 3
+ (hm1 + b 2 4 In ( I h l ) where a l = 0.4630151
bl
=
0.2452727
a2 = 0.1077812
b2
=
0.0412496
*
(V+
-
-
T)=0
(17)
where ri is the local interfacial normal.
where
a2 = 0.0725296
Thus T is computed everywhere in the conducting tissue a priori. It is clear that the right-hand side of (13) is identically zero if this assignment is adopted for T. The problem degenerates to solving Laplace’s equation for 9 subject to the boundary condition: ri
. (bo + b l m l + b2m:) In ( l / m l )
E,(m) = (1
For circular coils the magnetic vector potential is also easily expressed in terms of elliptic integrals
111. RESULTS The problem was worked out for the geometry in Fig. 1,for both the current vector potential approach and the T - 9 approach. Shown in Fig. 2 is the vector potential solution of the induced electric field. The contour plots in Fig. 3 represent the equipotential F surfaces which are also the current lines. A 3-D picture of the current vector potential is shown in Fig. 4, the height repre2enting the magnitude intensity of the vector potential A . For this problem, the potential only has a z component. By+comparison the induced electric field predicted by the T method is shown in Fig. 5. The two methods differ on average by 5 % . By way of giving the reader a quantitative index, Fig. 6 shows the induced vertical electric field along the horizontal midline of the tissue medium (at y = 2 in.).
+
(1 1)
and
m,=l-m K(k) = Ko(k2)
E(k) = Eo(k2).
B. Vector Plus Scalar Representation of the E Field The second method is appropriate to 2-D or 3-D problems, lending itself more readily to other orientations of the source coil. We begin by representing the electric field as the sum of another vector plus a scalar,
E
=
T
-
v+.
(12)
IV. CONSIDERATIONS I N INHOMOGENEOUS MEDIA The techniques have been tailored to the case where the conductivity has no spatial dependence. When this assumption is invalid, the gradient of conductivity (or resistivity) enters i;to the formulations for both the vector potential and the T - 9 method. With the vector pqtential approach the defining equation is (5). For the T - 9 method we must re+turn to (12). Enforcing the divergence free condition on J gives
v29 + v9 . Vu = V .
The divergence free condition on J dictates that
-
U
v2+ = v T. Faraday’s law requires
(13)
as-
+
V X T = -at
It follows that T can be related to the magnetic vector potential directly, i.e.,
-
T =
2
aA
--
at
-
Vu T S - * U
-
T.
(18)
The reader is justified in asking the question “Of what advantage is the technique over a conventional finite element approach?” Remember that any numerical technique can be employed to solve (5) or (18) including finite difference, finite element, or boundary integral approaches. In the 3-D inhomogeneous problem, no advantage is gained with the vector potential formulation, but a significant gain is realized with the T - method. Even though the conductivity gradient must be added to the numerical discretization, the problem is still scalar. The
+
42 I
DAVEY et al.: MAGNETICALLY INDUCED ELECTRIC FIELDS I N BIOLOGICAL TISSUE J = Curl Ar
1 ' Note: The length of the left bottom corner vector is 8.47 Vim.
1 --
--
f
/ / / / -
/
/
X
Y
\
f
-
Fig. 2. Induced electric field in the tissue as obtained by the current vector potential method.
Induced Current
Fig. 4 . Three-dimensional vector potential contour plot
10
A
A
4
0 LL
9 -10
A
+
--
-30
0
2
4
6
8
10
Distance (cml
Fig. 5 . Induced vertical field on the mid horizontal line of Fig. 1 for both methods 1 (vector potential A ) and method 2 ( T - V%). E = T-Gradient Phi
Induced Current Fig. 3 . Current vector potential plot in the conducting tissue region. The contour lines represent current paths.
problem is r5duced from solving three problems to only one. Again T is gotten effectively for free directly from the elliptic integral formulation for A,. V . CONCLUSION Noninvasive magnetic stimulation of neurons in the brain is realized by high intensity rapidly changing magnetic fields. The two methods discussed for predicting the induced fields capitalize on efficient means of incorporating the source field into the equations via elliptic integrals. Easy visualization of the internal currents is realized with the current vector potential since the vector potential surfaces are themselves current trajectories. The T - 9 method allows for complete solution of the vector field using only scalar quantities in any coordinate system.
0 Note: The length of the left bottom corner vector is 7.8 Vim.
Fig. 6 . Induced electric field as predicted by representing the electric field as the sum of a vector plus the gradient of a scalar.
ACKNOWLEDGMENT The authors wish to acknowledge the support of the Veterans Administration for their support _ _ of this work.
422
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 5, MAY 1991
REFERENCES [ I ] A. T. Barker, I. L. Freeston, R. Jalinous, and J . A. Jarrat, “Magnetic stimulation in clinical practice,” in Non-invasive Stimulation ofBrain and Spinal Cord: Fundamentals and Clinical Applicafions, P. L. Rossini and C . D. Marsden, Eds. New York: Alan Liss, 1988, pp. 231241. [2] D. Clauss, A. E. Harding, C . W. Hess. K. R. Mills, N. M. Murray, and P. K. Thomas, “Central motor conduction in degenerative ataxic disorders: A magnetic stimulation study,” J . Neurology Neurosurg. Psychiatry, vol. 51, pp. 790-795, 1988. [3] C. W. Hess, K. R. Mills, N. M. Murray, and T . N. Schriefer, “Magnetic brain stimulation: Central motor conduction studies in multiple sclerosis,” Ann. Neurology, vol. 22, pp. 744-752, 1987. [4] D. A. Ingram, A. J. Thompson, and M. Swash, “Central motor conduction in multiple sclerosis: Evaluation of abnormalities revealed by transcutaneous magnetic stimulation of the brain,” J . Neurology Neurosurg. Psychiatry. vol. 51. pp. 487-494, 1988. [SI V. E. Amassian, J. B. Cracco. R. Q. Cracco. L. Eberle. P. J . Maccabee, and A. Rudell, “Suppression of human visual perception with the magnetic coil over occipital cortex,” J . P h y s . , vol. 398, pp. I 40, 1988. [6] A. T. Barker, R. Jalinous, and I . L. Freeston. “Non-invasive magnetic stimulation of human motor cortex,” Lancet, vol. i , pp. 11061107, 1985. 171 P. A. Merton and H. B. Morton, “A magnetic stimulator for the human motor cortex,” J . Phys., vol. 381. 1986. [8] L. A. Geddes, “Optimal stimulus duration for extracranial cortical stimulation,”Neurosurgery. vol. 20, pp. 74-93, 1987. [9] C. M. Epstein, D. G. Schwartzberg, K. R. Davey. and D. B . Sudderth, “Localizing the site of magnetic brain stimulation in man,” Neurology, in press. [IO] V. E. Amassian, M . Stewart, G . J . Quirk, and J. L. Rosenthal. “Physiological basis of motor effects of a transient stimulus to cerebral cortex,” Neurosurgery. vol. 20, pp. 74-93. 1987. 1111 K. R. Foster and H . P. Schwan, “Dielectric properties of tissues,” Handbook of Biological Effects of Electromagnetic Fields, C. Polk and E. Postow. Eds. Boca Raton, FL: CRC Press, 1986, pp. 2896. [ 121 K. R. Davey, C:H. Cheng, and C. M. Epstein, “An alloy-core stimulator for magnetic brain stimulation,” presented at IEEE Conf. Biomed. Eng., New Orleans, LA, Nov. 1989. [ 131 J. Melcher, Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. 1141 S . Rush, J. Abildskov, and R . McFee, “Resistivity of body tissues at low frequencies.” Circulafion Res., vol. 12, pp. 40-50, Jan. 1963. [15] W . Kaune and M . Gillis, “General properties of the interaction between animals and ELF electric fields.’’ Bioelectromug., vol. 2. pp. 1-11, 1980. [I61 A. Chiba, K. Isaka, M. Kitagawa, Y. Yokoj. M. Nagata, and T. Matsuo, “Application of finite element method to analysis of induced densities inside human model exposed to 60 Hz electric field,” IEEE Power App. Syst., vol. PAS-103, pp. 1895-1902, July 1984. (171 D. Deno, “Currents induced in the body by high voltage transmission line electric field-Measurement and calculation of distribution and dose.” IEEEPower App. Sysf.. vol. PAS-96. no. 5 . pp. 1517-1527. Sept. 1977. [I81 W. Kaune and F. MacCreary, “Numerical calculation and measurement of 60-Hz current densities induced in an upright cylinder,” Biomag., vol. 6, pp. 209-220, 1985. 1191 R. Spiegel, “A review of numerical models for predicting the energy deposition and resultant thermal response of humans exposed to electromagnetic fields,’’ IEEE Trans. Microwave Theory Tech., vol. MTT-32, pp. 730-746, Aug. 1984.
1201 R. Speigel, “Numerical determination of induced currents in humans and baboons exposed to 60 Hz electric fields,” IEEE Trans. Elecfromag. Compat., vol. EMC-23. pp. 382-390, Nov. 1981. [21] R. Smythe, Static and Dynamic Electricity. New York: McGraw Hill, 1965, pp. 290-291. 1221 Abramowitz and Stegun, Handbook of Mathematical Functions, ninth ed. New York: Dover, 1979, pp. 591-592.
Kent R. Davey (S973-M’76-M’85-SM’89) was born in New Orleans, LA, on September 29, 1952. He received the B.S. degree in electrical engineering from Tulane University, New Orleans, in 1974. He received the M.S. degrees in power engineering and in physics, from Carnegie Mellon University, Pittsburgh, PA, and from the University of Pittsburgh. respectively, both in 1976. He received the Ph.D. degree in the field of continuum electromechanics at the Massachusetts Institute of Technolonv. Cambridge. MA. in 1980. He has been an Assistant Professor in Electrical Engineering at the Georgia Institute of Technology, Atlanta, since 1980. From 1979 to 1980. he worked as an Assistant Professor at Texas A&M University, College Station, TX. From 1974 to 1976, he worked as a development engineer at Westinghouse in electric motor design. His research interests are in predictions of magnetic field transients, electromagnetic scattering, and bioelectric phenomena analyses. -
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38. NO. 5. MAY I Y Y I
Prediction of Magnetically Induced Electric Fields in Biological Tissue Kent R. Davey, Senior Member, IEEE, Chin Huei Cheng, and Charles M. Epstein
Abstract-There are many potential medical applications in which it is desirable to noninvasively induce electric fields. One such application that serves as the backdrop of this work is that of stimulating neurons in the brain. The magnetic fields necessary must be quite high in magnitude, and fluctuate rapidly in time to induce the internal electric fields necessary for stimulation. Attention is focused on the calculation of the induced electric fields commensurate with rapidly changing magnetic fields in biological tissue. The problem is not a true eddy current problem in that the magnetic fields induced do not influence the source fields. Two techniques are introduced for numerically predicting the fields, each employing a different gauge for the potentials used to represent the electric field. The first method employs a current vector potential lanal_ogous to A in classical magnetic field theory where V X A = B ) and is best suited to two-dimensional (2-D) models. The second represents the electric field as the sum of a vector plus the gradient of a scalar field; because the vector can be determined quickly using Biot Savart (which for circular coils degenerates to an efficient evaluation employing elliptic integrals), the numerical model is a scalar problem even in the most complicated three dimensional geometry. These two models are solved for the case of a circular current carrying coil near a conducting body with sharp corners.
I. INTRODUCTION/BACKGROUND T PRESENT, extracranial magnetic stimulation is used to study human motor systems [1]-141; preliminary research is directed towards visual and cognitive physiology [5]. Typical magnetic stimulators produce peak currents of several thousand amperes in brief pulses lasting 50-200 ps. Such currents are produced by capacitor discharge at 700-4000 V [6]-[8]. The energy per stimulus is high, and the efficiency of coupling to the brain is low The locus of cerebral motor stimulation appears to lie near the junction of white and gray matter, at least 4-5 mm below the cortical surface and 17 mm beneath the scalp [9], [lo]. There has been no simple technique for predicting the induced electric fields. Without detailed analytical calculations of the field, it is difficult to charac-
A
Manuscript received November 14, 1989; revised June 1 1 , 1990. This work was supported in part by a grant from the Emory-Georgia Tech Biomedical Research Foundation, in part by the Georgia Institute of Technology, Emory University, School of Medicine, and in part by the Rehabilitation Research and Development Center of the Atlanta VAMC. K . R. Davey and C . H . Cheng are with the School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332. C. M . Epstein is with the Department of Neurology, School of Medicine, Emory University, Atlanta, GA 30322. IEEE Log Number 9144692.
terize the strength and orientation of the neuronal stimulus, or to optimize the design of the stimulus coils. The ultimate objective is to produce a magnetic stimulator that can activate the cortex at high frequency, and thus facilitate applications in cognitive physiology [ 1 11. The fields so generated must be accurately quantified by position. Presented here is an alternative analysis approach to the conventional finite element procedure {hat seeks a threedimensiota1 (3-D) forqulation of the H field or potential 9 where H = -V(+); E is then typically realized through the curl of the magnetic field. Thus, the principal contribution of this work is the presentation of a fast, accurate, and efficient technique for solving the internal electric field induced by a rapidly changing magnetic field. In this derivation we assume a homogeneous, mildly conducting medium, ignoring the presence of the skull. This simplification is justified because the conductivity of the bone is much lower than that of the brain; the side of stimulation appears to lie beneath the brain surface [9], [lo], and there is no physiological reason to believe that the local currents which stimulate neurons extend appreciably outside the cortex. Magnetic field stimulation of the brain (soft tissue) is characterized by the following: characteristic length I = 0.1 m permeability y = 4.rr . IO-’ H/m dielectric constant E = 10‘ = 1/36.rr . 10’ F/m conductivity U = 0.06 S/m characteristic system time 7 = l/frequency = 0.1 ms electrical relaxation time 7, = € / U = 160 ys magnetic diffusion time 7, * = 0.7 ns = 300 ns electromagnetic time 7,,,,= quasistatic parameter p = {7,,,/7}’ = 1.0 x
x;
As shown by Melcher [13], a system with the aforementioned characteristic times is magnetoquasi-static and is represented by the following equations:
v
x H =
V
X
%E 7
+
E
=
+p
g]
ai
--.
at
at the frequencies characterizing The magnitude of magnetic field stimulation is sufficiently small that the second term on the right of (1) can be ignored without any
0018-9294/91/0500-0418$01.00 0 1991 IEEE
DAVEY et a!.: MAGNETICALLY INDUCED ELECTRIC FIELDS IN BIOLOGlCAL TISSUE
loss of accuracy. The value of will be even smaller at higher frequencies due to the decrease of the dielectric constant [ 141, [ 151. By taking the divergence of (1) and recalling the fact that the divergence of the curl of a vector is zero, the equation of interest in the tissue reduces to
v
*
OB =
v . 7 = 0.
11. ELECTRIC FIELDPREDICTION We will discuss two methods for predicting the induced electric fields from the time changing B fields. The first employs a current vector potential and is especially suited to two dimensional simulations. In three dimensions this technique would necessitate a vector field solution with a large number of unknowns. The second technique represents the electric field as another vector plus the gradient of a scalar. The vector part is found and expressed immediately in terms of the magnetic vector potential of the source coil. The scalar satisfies Laplace’s equation and is determined to insure that the boundary conditions on the current density are realized. This method is suitable for 2-D and 3-D problems, and always involves scalars only.
A. Current Vector Potential Approach We begin this procedure by representing the current density as the curl of a vector
7-vxF
1 cm
/
1 in dia Coil
(3)
Calculations of induced current densities have been approached a number of ways to date. Chiba er al. [16] attack the problem via a finite-element analysis involving the minimization of an energy functional. Because they are concerned about 60 Hz transmission line electric field effects, they do not use magnetoquasistatics; their primary focus is the external electric field of the line. Like the work by Den0 and Kaune [17], [18], they examine surface charge effects and the internal currents accompanying changes of these surface charges. Considerable attention must be given to exposed tissue surface areas, an emphasis that is peripheral to the problem as viewed here in its eddy current context. Spiegel [19], [20] attempt a solution of the complete Maxwell Equations (even at 60 Hz) in terms of the incident and scattered field.
(4)
With this assignment we are guaranteed a divergence free current density. In terms of the resistivity p = l / a , (2) can be written as
The magnetic field density B is caused solely by the source currents in the excitation coil. Fig. 1 shows a test geometry for the induced field prediction. In practice the coil is frequently placed flat against the heat; this geometry serves to better delineate the differences between the two techniques under discussion in this paper. Note that we have deliberately chosen a square rather
-+ I 419
Conducting Medium conductivity 1 Slm
Source Current 10 t u r n s ramp up in 5 0 p s
-
4 in
/ In
Fig. 1 . Test geometry for the determination of the induced electric field generated by a coil ramping up to 31 536 amp turns in 50 ps. The conductivity of the tissue is taken here to be I S/m. The square cross-section conducting medium is assumed to be long in extent with the same crosssection for all z . The excitation coil is a simple current loop existing only a t z = 0.
than circular geometry as an exceptionally rigorous test of the techniques. The square corners contort the current paths, and thus tax the numerical model more than a cylindrical or spherical model would. In a homogeneous conducting region, (5) reduces to
-
V2F =
ai
0-.
at
This equation may be discretized using either a finite difference or finite-element procedure. We employed a finite difference impLementation of (6) and enforce the condition that ri llJll = 0, i.e., that no normal current exists on the tissue air interface. I ) Source Coil Field Prediction-Biot-Savart: The principal issue is how to calculate the main source field that drives ( 5 ) . The generalized Biot-Savart rule involves the integration of the source current:
(7) Although this formulation always works, it can be time consuming. Recall that for either a finite difference or finite-element implementation, the source term must be evaluated at every node point in the grid. In three dimensions the overhead becomes intolerable. 2) Source Coil Field Predictions-Elliptic Integrals: The fields for circular coils are more easily solved using elliptic integrals. Smythe [21] shows that for a coil of radius a carrying I amps, the magnetic field density at some radial displacement p and axial position z above the coil is
. j-K(k) +
+
a2 p2 + z 2 (a - p)2 + z 2
420
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38. NO. 5. MAY 1991
(9) where k2
=
m=
4a
P
(a
+ p)? + z 2 .
1 - J 1 - k2 1
+ m’
Abramowitz and Stegun [221 provide an nition of elliptic integrals that are easily evaluated in terms of polynomial approximations:
Ko(m) = (ao
+ a l m l + u2m3
=
1.3862944
bo
=
0.5
U! =
0.11 19723
61
=
0.1213478
62 = 0.0288729
(10)
+ alml + a 2 m 3
+ (hm1 + b 2 4 In ( I h l ) where a l = 0.4630151
bl
=
0.2452727
a2 = 0.1077812
b2
=
0.0412496
*
(V+
-
-
T)=0
(17)
where ri is the local interfacial normal.
where
a2 = 0.0725296
Thus T is computed everywhere in the conducting tissue a priori. It is clear that the right-hand side of (13) is identically zero if this assignment is adopted for T. The problem degenerates to solving Laplace’s equation for 9 subject to the boundary condition: ri
. (bo + b l m l + b2m:) In ( l / m l )
E,(m) = (1
For circular coils the magnetic vector potential is also easily expressed in terms of elliptic integrals
111. RESULTS The problem was worked out for the geometry in Fig. 1,for both the current vector potential approach and the T - 9 approach. Shown in Fig. 2 is the vector potential solution of the induced electric field. The contour plots in Fig. 3 represent the equipotential F surfaces which are also the current lines. A 3-D picture of the current vector potential is shown in Fig. 4, the height repre2enting the magnitude intensity of the vector potential A . For this problem, the potential only has a z component. By+comparison the induced electric field predicted by the T method is shown in Fig. 5. The two methods differ on average by 5 % . By way of giving the reader a quantitative index, Fig. 6 shows the induced vertical electric field along the horizontal midline of the tissue medium (at y = 2 in.).
+
(1 1)
and
m,=l-m K(k) = Ko(k2)
E(k) = Eo(k2).
B. Vector Plus Scalar Representation of the E Field The second method is appropriate to 2-D or 3-D problems, lending itself more readily to other orientations of the source coil. We begin by representing the electric field as the sum of another vector plus a scalar,
E
=
T
-
v+.
(12)
IV. CONSIDERATIONS I N INHOMOGENEOUS MEDIA The techniques have been tailored to the case where the conductivity has no spatial dependence. When this assumption is invalid, the gradient of conductivity (or resistivity) enters i;to the formulations for both the vector potential and the T - 9 method. With the vector pqtential approach the defining equation is (5). For the T - 9 method we must re+turn to (12). Enforcing the divergence free condition on J gives
v29 + v9 . Vu = V .
The divergence free condition on J dictates that
-
U
v2+ = v T. Faraday’s law requires
(13)
as-
+
V X T = -at
It follows that T can be related to the magnetic vector potential directly, i.e.,
-
T =
2
aA
--
at
-
Vu T S - * U
-
T.
(18)
The reader is justified in asking the question “Of what advantage is the technique over a conventional finite element approach?” Remember that any numerical technique can be employed to solve (5) or (18) including finite difference, finite element, or boundary integral approaches. In the 3-D inhomogeneous problem, no advantage is gained with the vector potential formulation, but a significant gain is realized with the T - method. Even though the conductivity gradient must be added to the numerical discretization, the problem is still scalar. The
+
42 I
DAVEY et al.: MAGNETICALLY INDUCED ELECTRIC FIELDS I N BIOLOGICAL TISSUE J = Curl Ar
1 ' Note: The length of the left bottom corner vector is 8.47 Vim.
1 --
--
f
/ / / / -
/
/
X
Y
\
f
-
Fig. 2. Induced electric field in the tissue as obtained by the current vector potential method.
Induced Current
Fig. 4 . Three-dimensional vector potential contour plot
10
A
A
4
0 LL
9 -10
A
+
--
-30
0
2
4
6
8
10
Distance (cml
Fig. 5 . Induced vertical field on the mid horizontal line of Fig. 1 for both methods 1 (vector potential A ) and method 2 ( T - V%). E = T-Gradient Phi
Induced Current Fig. 3 . Current vector potential plot in the conducting tissue region. The contour lines represent current paths.
problem is r5duced from solving three problems to only one. Again T is gotten effectively for free directly from the elliptic integral formulation for A,. V . CONCLUSION Noninvasive magnetic stimulation of neurons in the brain is realized by high intensity rapidly changing magnetic fields. The two methods discussed for predicting the induced fields capitalize on efficient means of incorporating the source field into the equations via elliptic integrals. Easy visualization of the internal currents is realized with the current vector potential since the vector potential surfaces are themselves current trajectories. The T - 9 method allows for complete solution of the vector field using only scalar quantities in any coordinate system.
0 Note: The length of the left bottom corner vector is 7.8 Vim.
Fig. 6 . Induced electric field as predicted by representing the electric field as the sum of a vector plus the gradient of a scalar.
ACKNOWLEDGMENT The authors wish to acknowledge the support of the Veterans Administration for their support _ _ of this work.
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 5, MAY 1991
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1201 R. Speigel, “Numerical determination of induced currents in humans and baboons exposed to 60 Hz electric fields,” IEEE Trans. Elecfromag. Compat., vol. EMC-23. pp. 382-390, Nov. 1981. [21] R. Smythe, Static and Dynamic Electricity. New York: McGraw Hill, 1965, pp. 290-291. 1221 Abramowitz and Stegun, Handbook of Mathematical Functions, ninth ed. New York: Dover, 1979, pp. 591-592.
Kent R. Davey (S973-M’76-M’85-SM’89) was born in New Orleans, LA, on September 29, 1952. He received the B.S. degree in electrical engineering from Tulane University, New Orleans, in 1974. He received the M.S. degrees in power engineering and in physics, from Carnegie Mellon University, Pittsburgh, PA, and from the University of Pittsburgh. respectively, both in 1976. He received the Ph.D. degree in the field of continuum electromechanics at the Massachusetts Institute of Technolonv. Cambridge. MA. in 1980. He has been an Assistant Professor in Electrical Engineering at the Georgia Institute of Technology, Atlanta, since 1980. From 1979 to 1980. he worked as an Assistant Professor at Texas A&M University, College Station, TX. From 1974 to 1976, he worked as a development engineer at Westinghouse in electric motor design. His research interests are in predictions of magnetic field transients, electromagnetic scattering, and bioelectric phenomena analyses. -
