The effect of ohmic return currents on biomagnetic fields

J. theor. Biol. (1987) 125, 187-191

The Effect of Ohmic Return Currents on Biomagnetic Fields JOHN PAUL BARACH

Department of Physics, Vanderbilt University, Nashville, TN 37235, U.S.A. (Received 7 April 1986, and in revised form 21 October 1986) To illustrate the calculation methods used for biomagnetic fields we present a detailed calculation of the B-field from a spherical cell in an infinite ohmic bath. The calculation is done from three approaches and the results are used to clarify some misinterpretations that may seem to be biophysical problems but are, in fact, creatures of the formalism used. Many workers are now interested in the magnetic fields generated some relatively large distance away from a biological source current such as that in a fiber action potential. Examples are Plonsey (1981), Kaufman & Williamson (1982) and Romani & Leoni (1984). The magnetic fields are a measure of the impressed currents but the measured fields are reduced by "return currents" within the medium and interior to the measuring distance. This point can be treated in various ways (Barach et al., 1985; Roth & Wikswo, 1985) but it was felt that it might be useful here to set out clearly the equivalence of three of the principal ways of dealing with biomagnetic field calculations. Consider a spatially limited source of impressed current at the origin in an infinite ohmic medium (e.g. a strip of cardiac fiber in a saline bath) and consider measuring the magnetic field out at some radius, R considerably larger than the source radius. There are three ways of stating the production of field, B(R). (i) Deduce the B-field from the law of Biot-Savart in which case all currents J throughout the medium must be added up, even those external to R or at large axial positions away from the measuring point. (ii) Use Ampere's circuital theorem to calculate B(R). In this case, no currents external to the circuit around the source at radius R contribute, the field is due to the internal impressed currents less such return currents as circulate within R. (iii) Notice that a Green's function solution for B(R) contains only Curl J as a source term so neither the Ohmic return currents nor the uniform impressed current density produces magnetic field, it is a result only of the region of Curl J at the membrane. Clearly, the three approaches above are equivalent and give the same result but they produce quite different mental pictures as guides to analysis. Formal proofs of the equivalence of (i)-(iii) can fail to be helpful in interpreting actual magnetic data so we present a particular calculation here in which this equivalence can be demonstrated explicitly and with unusual clarity. We calculate the magnetic field produced by an ideal small spherical cell at the origin, the cell having impressed currents, which are, at some instant of time constant and uniform in the axial direction. The cell and infinite bath are of uniform, scalar conductivity. Everywhere the permeability is that of free space,/Zo, and the impressed current inside the cell 187 0022-5193/87/060187+05 $03.00/0

© 1987 Academic Press Inc. (London) Ltd

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J.P. BARACH

is, by construction, uniform J = Joti~. This problem has the advantage of having a well-known (Reitz et al., 1980) solution: that of the uniformly magnetized or polarized sphere. We note that the solution for the J-field is that of the D-field of a uniformly polarized sphere and that in both the inside (where it is uniform) and outside regions, J has neither divergence (because of the continuity equation in steady conditions) nor curl (because E is curl-free and J is ohmic). So in this problem we know J everywhere .~. = 2/3Jo(cos 0~ir-sin 0d0) Jout = 1/3Joa3( 2 cos 0tit + sin

Oao)/r 3.

This solution is basically the textbook solution for a small spherical dipole, with lines running uniformly upward within and curving around and back outside. It has zero divergence within and without, of course, but also on the boundary at r = a across which Jr is continuous, as may be seen from inspection. The curl of J is zero within and without, also, but on the boundary the Jo component changes by Jo sin 0 in the thin m e m b r a n e of thickness d. Now curl /~ =/Zo.~ and if one takes the curl of that equation and remembers that div/~ = 0, one obtains a Poisson equation for/~ V2/~ = -/XoV x j which has the solution, for bounded current fields which vanish at infinity

~(,~)

f vIR-x J ('1h

~._~o

-47r

d3r"

(1)

This may be argued directly from an analogy with the electric case or obtained from the law of Biot-Savart by utilizing the fact that g r a d ( l / R - r) = (/~ - F)/IR - rl 3. One must then expand the cross-product in that law and discard the term which is an integral of curl ( J ( r ) / I R - rl) over all space since it may be expressed by a Stokes integral of the integrand at a boundary taken to be very distant. We study now the three different ways of calculating the B-field: Ampere's Circuital Law for the current enclosed, method (ii), which is basically equivalent to the scalar potential method. The Green's function approach, in which merely the Curl J(r) (not J(r) itself) is the source of the field, is (iii) above. Lastly, we will do method (i) and utilize the law of Biot-Savart or the vector potential, which is its logical equivalent and treat all the currents as sources of B-field. For a case in which the distinctions can be drawn, the calculations tend to be very hard, but we try now to elucidate the case at hand. We will limit ourselves to a calculation of B ( R ) where the field point, P, is at radius R in the x - y (equatorial) plane. The calculation is tedious enough even for this case. First we utilize Ampere's law in the equatorial plane. B must be in the ~b-direction by symmetry; at point P, that is the y-direction (see Fig. 1 below) if its location = Rtgx. (We locate the currents by spherical r, 0, ~b coordinates which must be kept distinct from the x, y, z coordinates of the field point.) The total current enclosed is the uniform internal current 7ra2Jo2/3 less the "return currents" between

O H M I C R E T U R N C U R R E N T S AND B I O M A G N E T I S M

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the spherical source and the radius R: These are given by

Joa3/3 I f 2"trr r3 dr = 2"rrJoa2/3( a/ R - 1). Thus

B( R ) = lXoJoa3/3 R2 ~v.

(2)

In this approach we have a B-field determined by the currents within R only. The field dependence is more steep with distance than the R -~ of a wire because of return current cancellation which is included at larger R. Case (iii): We utilize the Green's function approach and identify the regions of Curl J as the sources of B-field. Since the J-field has curl only in the membrane at radius a (infinitesimal thickness d) the volume integral in (1) becomes a surface integral at r = a of Curl J multiplied by d: that is just J0 sin 0 in the d,b direction. Now the direction of ~6 varies with 4, and inspection of Fig. 1 will show that the two halves a t + a n d - ~ b cancel so that only the transverse, ~,, component at P remains. This inserts a cos ~b into the integral. Then in Fig. 1: b = a cos 0, c = ( R 2 + a 2 s i n 0 - 2 a R s i n 0 c o s ~ b ) 1/2, d = ( R 2 + a 2 - 2 a R s i n O c o s q S ) ~/2, e = a s i n 0 . So that ___/ZoJo d,, I o " f ~ ~"

B(R)-

4~

a2sin2Oc°s4) dOdck (R2+a2-2aRsinOcosc~)l/2"

"

(3)

Integrals of this hypergeometric form are generally recalcitrant but it is important to show how, not just the fact that, the different approaches to the calculation give the same result. To do this manageably we approximate and expand the radical binomially so that: ( l + x ) -t/2 = t-~_x±~x " l --3 2 -~6x 5 3--35 tr~_sx 4 - ~ 63 x 5 in which we factor out an R and let x = a 2 / R 2 - 2 a / R sin 0 cos ~b. The th integration is over 2~ and

,[

°S 8

I

*

~

p

R

•

]

FiG. 1. Geometry of the calculation. The cell has radius a. The other letters in lower case denote distances used in the calculation. The field point is P, at radius R, and the cell current is uniformly in the upward z-direction.

190

J.P. BARACH

SO only terms of even powers o f cos $ are non-zero. If we keep terms up to ( a / R ) 5 then the integrand b e c o m e s

a / R sin 3 0 cos 2 $ + ( a / R ) 3 ( 5 sin 5 0 cos 4 $ - 3 s in3 0 cos 2 $ ) + ( a / R ) S ( ~ s i n 3 0 cos 2 $ - ~ s i n

5 0 cos 4 $ + ~ s i n

7 0 cos 6 $ ) .

The integral over 2zr in $ and ~r in 0 then yields = P'°j°a2 ayar R ( ] + (a/R)2(2 -~6) + ( a / R ) 4 ( 5 - ~

4~rR

+ 26~))

B( R ) - lz°j°a3 d~, 4 4R 2 3" This is the same result as (2) but now it a p p e a r s as if neither of the O h m i c regions p r o d u c e any of the B-field; the source is only the thin layer in which Curl J differs from zero! The faster than 1/R fall off with distance is not due to return currents at all. The field source is a localized region o f Curl J and the spatial d e p e n d e n c e comes, as from a dipole, from the increasingly better cancellation, as distance increases, o f the near, +ti,, portion, by the far, -d.,, portion of the source. Finally, we wish to use the law of Biot-Savart directly. The calculation is, however, tedious even in our a p p r o x i m a t i o n s . (We could use a vector potential f o r m a l i s m which also treats all the currents as field sources, but the second, Curl A = B, step would confuse our a p p r o x i m a t i o n . ) So here we write

tZo r J ( r ) x ( R - F)

B ( R ) = ~-~ -

]-~-rl T

d3r.

(4)

Referring to Fig. 1 again, we can indicate the contribution from the cell interior in a m a n a g e a b l e way but we will not pursue the calculation further out to the Jout currents as the matters b e c o m e s repetitious and the required remarks can be m a d e by calculating that portion o f B (call it B') due to the J~n. In Fig. 1 the IR - r[ 3 term needed in (4) is d 3= (R2+r2-2rR sin 0 cos 0) 3/2. The v o l u m e integral is now for r from 0 to a and brings in a factor of r 2 sin 0. The cross-product in (4) will have an x and y part, and, as before, the x portions cancel with the 0 integration and the y parts add. The y parts, then, are given by the scalar p r o d u c t o f J and the length f = R - (r sin 0 cos $ ) . R e m e m b e r i n g that ~n = ]J0d.., /~'(R) = - ~ - - tir

I~oJo

f o ' f o ~ f o '~r2

sin O(R - r sin 0 cos $ ) dr dO d $ (R2+ r 2 - 2 r R sin 0 cos $)3/2

The p r o c e d u r e is as before. We e x p a n d in powers of r/R, d r o p o d d powers of cos $ and p e r f o r m the integrals. As a guide to the algebra, the result after the $ integration is

.o,o for: r2(2 sin O+(r/R)2(~sin 3 0 - 3 )

/~'(R) = ~ - ? - ~i~

+(r/R)4(~-sin 0 - ~ s i n 3 0+~,5 sin 5 0)) dO dr.

OHMIC RETURN CURRENTS AND BIOMAGNET|SM

191

The ( r / R ) 4 term precisely cancels after the radial integration so that we are left with B'( R ) = tzogoa3~v/3R2(2 + 3 ( a / R)2 + . . . ). That is, this approximate Biot-Savart calculation, utilizing just the inner uniform currents, gives the correct result to 0.967 at R = a in this fifth order approximation. Since Ampere's Law teaches us that currents external to point P cancel out in their effect, this seems reasonable. However, even at very large R, the inner current alone via the Biot-Savart calculation gives 70% of the correct field. We understand this by noting that the outer Ohmic currents have portions along the z-axis region which aid the field and opposing portions in the equatorial regions. That the calculation is so close to correct ignoring Jou, entirely is a function both of the widespread and of the largely cancelling nature of the outside currents. So we might say from the Biot-Savart point of view, even incompletely carried forward, that the B-field is largely produced by the interior Ohmic source currents and the return currents almost cancel out. The spatial fall-off with R is due less to dipolar geometrical considerations than to the 1/r 2 force law for a current element. Indeed, we now join elementary texts which present Biot-Savart so as to resemble the Coulomb 1/r 2 dependence. That is, we here almost rediscover the cell's impressed current as an isolated Idl field source whose return currents are so often ignored in elementary presentations, and from which the field drops as l/r-'. In summary, one notes that, of course, the three methods all work. They, however, lead to different mental pictures. If one is measuring fields from action potentials within a nerve bundle, the argument that it is essential to account for return currents within the bundle is true within formalism (ii). A worker utilizing e.g. method (iii) will be equally correct in ignoring such currents altogether. For another example, a calculation of type (i) can quite safely ignore the actual membrane currents and assert they "contribute negligibly to the B-field" (their volume integral in (4) is very small) whereas it is only the membrane currents which give Curl J, and all other currents can be ignored in formalism (ii). Finally, one sees that the same sort of arguments can confuse distinctions as to the multipolarity of the field source. It is therefore essential to be clear which field source formalism one uses in discussing biomagnetic calculations, a discipline which is unneccessary when currents are confined to run along wires in a simple way. The author thanks J. P. Wikswo, Jr and B. J. Roth for helpful conversations. This research was supported by the Office of Naval Research under contract N00014-K-0107 and by the National Institutes of Health under Grant l-R01 NS 19794-01. REFERENCES BARACH, J. P., ROTH, B. J. & WIKSWO, J. P., JR (1985). IEEE Trans. Biomed. Eng. BME-32, 136. KAUFMAN, L. & WILLIAMSON, S. J. (1982). Ann. N. Y. Acad. Sci. 388, 197. PLONSEY, R. (1981). Med. Biol. Eng. Comp. 19, 311. ROMANI, G. L. & LEONI, R. (1984). Biomagnetism: applications and theory. (WEINBERG, H., STROINK, G. & KATILA, T. eds). p. 205. Oxford: Pergamon Press. ROTH, B. J. & WIKSWO, J. P. JR (1985). Biophys. J. 48, 93. REITZ, J. R., MILFORD, F. J. & CHRISTY, R. W. (1980). Foundations of electromagnetic theory, 3rd edn. p. 207. Reading, Mass: Addison-Wesley.