Eccentric spheres models of the head

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 38, NO. 9. SEPTEMBER 1991 87 1

Eccentric Spheres Models of the Head B. Neil Cuffin

Abstract-Equations are derived for electric potentials (electroencephalograms) and magnetic fields (magnetoencephalo- grams) produced by dipolar sources in three eccentric spheres models of the head. In these models, I) the thickness of the layer representing the skull varies around the model, 11) the thickness of the scalp layer varies, and 111) the electrical conductivity of an eccentric spherical “bubble” in the brain region varies. Using these equations, it was found that variations in these features of the models have at most only small effects on the general spatial patterns of the electric potentials and the radial component of the magnetic fields. However, some significant effects on the amplitudes were found. The effects of the variations in the skull and scalp layer thicknesses on the field amplitudes were found to be significantly smaller than on the potential amplitudes. The effects on the field amplitudes of the variations in the bubble conductivity were found to be only somewhat smaller than on the potential amplitudes. It was also found that the effects of variations in these features of the models on source localization accuracy were significantly smaller for inverse solutions using fields than for solutions using potentials.

INTRODUCTION PHERICAL MODELS of the head are often used in S analyzing electroencephalograms (EEG’s) and mag-

netoencephalograms (MEG’s). For example, a spherical model is often used in inverse calculations to estimate the locations of sources in the brain which produce evoked EEG’s or MEG’s. For EEG’s, the spherical model usu- ally contains concentric layers with different electrical conductivities which represent the skull, scalp, and other tissues. For MEG’s, no such concentric layers are used since they have no effect on magnetic fields [ 11. However, there are differences between a spherical model and the head and these can cause errors in estimating the locations of sources. Many studies of the effects of various features of the head on EEG’s and MEG’s and source localization accuracy have been performed. While the effects of the general, nonspherical shape of the head have been studied in some detail, only one study [2] of the effects of variations in the thickness of the skull or scalp with position about the head has been performed. Results for only one set of model parameters are presented in that study. More studies probably have not been performed because only numerical computer methods [3] have been available for use in such studies. These models require large amounts of computer time and resources.

Manuscript received July 12, 1990; revised October 25. 1990. This work

The author is with the Francis Bitter National Magnet Laboratory, Mas-

IEEE Log Number 9101997.

was supported by NIH Grant NS22703.

sachusetts Institute of Technology, Cambridge. MA 02 139.

In this paper, equations are derived for the electric potentials (EEG’s) and magnetic fields (MEG’s) produced by dipolar sources in three eccentric spheres models in which I) the thickness of the layer representing the skull varies with position around the model, 11) the thickness of the scalp layer vanes, and 111) the electrical conductivity of an eccentric spherical “bubble” in the brain region of the model varies. The effects of these features of the models on the electric potentials and the radial component of the magnetic fields are determined by comparison with potentials and fields from a concentric spheres model. In addition, the effects of these features on source localization accuracy are determined by calculating inverse solutions in a concentric spheres model using potential and field data from the eccentric spheres models and compar- ing the solutions with the actual known sources.

DERIVATION OF EQUATIONS

Model I: Innermost Spherical Surface Shifted The general equations for the potentials produced by a

P, dipole in the three regions of the eccentric spheres model in Fig. l(a) are [4]

PI cos 4 v, = ~

4~02 n = ~

where PA and Pi are associated Legendre polynomials. Primed coordinates are used in regions 1 and 2 and unprimed coordinates in region 3. The origin of the primed coordinates is at z = U ; because of the spherical geometry, 4 = 4 ’ . In order to meet the boundary conditions at r = c , it is necessary to express V2 in unprimed coordinates. Using equations given by Morse and Feshbach [ 5 ] , V2 can be expressed in these coordinates as

P; (COS e) (s - I)! s + I

s = n

] (4) (-i)”-yn + I ) ! P ~ (COS e)

+ Dnun s = 1 (:I (n - s)!(l + s)!

and after interchanging the order of summation

0018-9294/91/0900-0871$01 .OO 0 1991 IEEE

a12 IEEE TRANSACTIONS ON BlOMEDlCAL ENGINEERING, VOL 38. NO. 9. SEPTEMBER 1991

(C) Fig. I. Eccentric spheres models. Model I is shown in (a). The dots on the surface of the model are the points used in the measurement grid for the potential inverse solutions; only one dimension of the grid is shown. For the magnetic field measurement grid, the points are I cm radially from the surface of the model. Model I 1 is shown in (b) and Model 111 in (c).

5 Pi (cos e) [ (;J+I (s - l)! P, cos +

v2 = _____ 4 T U 2 s = l

(S + l)! n = s (n - s)!

The constants B,, C,,, etc. can be evaluated by applying

(6)

the boundary conditions

VI = V2, r' = b

(7)

lIIlllIl1lllllllll llllll

Equations (6)-(10) can be solved to obtain the following equation for the Dn's:

n ( l - k1)b2"fID,7 5 n = ~ (nkl + n + l )a"+ ' (n - l ) ! ( ~ - n)!

5 un(- l ) f l -F(n + l>!D"

= - E (2n + l ) f " - '

(n - s)! n = T

S

r 7 = l (nk, + n + l ) a " + ' ( n - l)!(s - n)!

(1 1) where

C, , = k2[sR2'+' + ( s + l )c2 '+l]

C = R2S t I - c 2 S + 1 2s

and k l = uI /u2 and k2 = u2/u3. These D,'s can then be used in the following equation for the potential on the surface of the model:

P, cos + (2s + 1)2RSaS+ Is! P,' (cos e ) v, = ~ c 4nu3 s = 1 s2[C,, + (s + l)C2J

(2n + l)f"- ' + n ( l - k1)b2"+'D, n = l (nkl + n + l ) a "+ ' (n - l ) ! ( ~ - n)! . (12)

In practice, the infinite series in (1 1) is truncated at s,,, and smaxDn's are calculated from the s,,, simultaneous equations. These D,,'s are then used in (12) with s,,, being increased until there is no significant change in the value

The equations for the magnetic field B produced by a P, dipole are derived using the vector magnetic potential A from which

of v,.

B = V x A . (13) The equation for A is [6]

where J s is the current dipole moment per unit volume of the sources, U; and uJ' are the conductivities on the inside and outside of surface dsj, I/ is the potential on that surface, and

- - _

- m)! cos m(+f - + , ) P : : ( ~ ~ ~ e')~::(cos e,)

(15) (n + m)!

llul1llllllll Ill I II Ill I I I I II I I I Ill

CUFFIN: MODELS OF THE HEAD 873

where E , = 1 for m = 0 and E , = 2 for m # 0. The (1 - kl)bzn+lDIl coordinates with the s-subscript are at the source or on the

For this model, it is only necessary to evaluate the sur-

c = (nkl + + l ) a n + I ( ~ - - n)!

(CIS - sC2.J

spherical surfaces.

face integral in the second term in (14) over the surface of the shifted sphere. This is because the surfaces of the

- s(s + 1) ,sf! [CIS + (s + 1 ) C d [($ + 1YI2 (3

unshifted spheres produce only Bo and B, components [7]. a " ( - l ) " p S n ! D , Hence, it is only necessary to derive equations for the Bo.

the radial field component. Note that no B, or B,, com-

n = s (n - s)!

n = l (nkl + n + l ) a n + ' ( n - l)!(s - n)!

S (2n + 1)fn-I = - E (23)

and B,, components produced by the shifted sphere and then calculate their projections in the i direction to obtain

ponents are produced by the spherical surfaces [7]. and the potential on the surface of the model is given by The electric potential produced on the shifted surface

is given by P, (2s + i ) 2 ~ . y a s + + ' ( ~ - i)!p:(cos e) v 3 = - c P, cos 4 5 (2n + l)P;,(cOs e l ) 4au3 s = I [Cl, + (s + 1)C2Sl

. e (24)

v, = - 4TU, n = l [n(kl + 1 ) + 1]6"+' ' (2n + I ) f " - ' + (1 - kl )b2"f1Dn

n = l (kin + 12 + I ) u " + ' ( ~ - I)!(s - n ) ! ' - U - ' + bZn+l D,) (16) which along with (15) can be substituted into the second integral in (14). This integral can be evaluated using the orthogonality properties of the trigonometric functions and Legendre polynomials. After applying (13) to the evalu- Model Inner Two Surfaces shifted ated integral, the following equations for the field produced by the shifted surface are obtained:

Because Of the geometry, a pz dipo1e Pro- duces no magnetic fie1d.

The general expressions for the potentials produced by a P, dipole in regions 1 and 2 of the model shown in Fig.

The field produced by the dipole source alone can be calculated by evaluating the first integral in (14), or more directly, it can be calculated from the Biot-Savart law as

P,i x L 4aL3

B d = p -

where L is the vector from the dipole to the point at which the field is calculated. The projection of this field in the i direction is easily calculated.

The potential produced by a P; dipole can be derived in the same manner as for a P, dipole except that the general equations for the potential in the three regions of the model are

pz V3 = - c Ps (cos 0) 4au3 s = I

The equation from which the D,'s are calculated for a P, dipole is

~ ~

l(b) are the same as ( 1 ) and (2). In region 3, primed coordinates are also used

In order to meet the boundary condition at r = R , (25) is expressed in unprimed coordinates as

S

En * c n = ~ ~ " + ' ( n - I)!(s - n>!

(-1)"-"n + l ) ! a " F , (n - s)!

following equation for the F,I's is obtained:

c (S + l)! n = s

After applying appropriate boundary conditions, the

S

(C3n + C d F n nC2n + 1 c f l = I (csn f C6n)anf1(n - I)!($ - n)!

an( - l )" -s ( (n + l)!Flf

(2n + 1)2fn-1C2n+l = - e (27)

2 s + l m S2

-

[(s + 1)!]2 i:) 2 (n - s)! S

(c5n + C6n)anf1(n - l)!(s - n)!

874 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 38, NO. 9, SEPTEMBER 1991

where C3, = ( 1 - kl ) [ (n + l)k2 + n]b2"+I

c4, = ( 1 - kJ(nk1 + n + 1)c2'1+1

c,, = n(n + 1 ) ( 1 - k , ) ( l - k2)b2"+I

cfj, = (nk1 + n + I)(&* + n f l ) C 2 " + ' .

These F,'s are used in the following equation for the potential on the surface of the model

and the surface potential is given by 2 s + I

v3 = 5 5 (;) (2s + l ) ( s - l)!P:(CoS e)u"l 4nu3 s = I

Again, the magnetic field is zero for this dipole. _ _ P,cosu5 : (2s + l~s!P.!(cos8\ I J \ ~ -, v3 = _.

4nu3 ,f.I ( E ) S2 Model III: Eccentric SDhericul Bubble in the Bruin S (2n + 1 ) 2 f n - I 2 n + 1 + ,,2r1+1

x c ,=I (eK.. + C<-)a"+'(

\ ,

Equations for Bo, and B+r are derived in the same man- C A, PI , (cos 8' ) r"' (33) ner as for Model I except that now the surface integral in I/, = ~ 9, cos 4 OD

(14) must be evaluated for both shifted spherical surfaces. 4TUl , = I

P, cos These equations are

p P, sin 4 g P! (COS e' v2 = ___ 4KU2 n = I g (cos e l ) ( f& + B,7r'" + $), Bo, = ~

4 n sin e' ,= I (c~, + C6,)rm+ I

r ' < f (34) P, cos 4 OD

x {f"-'{[k,(nk, + n + 1 ) + (n + l)k2 v, = ___ c PI, (cos e r )

- (3n + 2)]cZn+I + (1 - k J ( 1 - k2) 4nU2 t i = I

- (3n + 2)]C2"+I + ( 1 - k , ) ( l - k2)

* (n + l )b2"+l} - C ~ " + I ( C ~ , ~ + C4n)F,i}. (30) For a P, dipole, the equation for the F,'s is

S

C 2 n + I ( C 3 n + C4,JFn c (c5, f C611)a"+'(n - l)!(s - n)!

s(s + 1) 2 s + 1 OD a"( - l )" -"n!F,

[(s + 1 ) ! 1 2 (5) n = s (n - s)! -

S (2n + 1)2fn-1C2n+l = - E (31) f l = I (c5, + C(jn)a"''(n - 1)!(s - n)!

The boundary conditions at r = c can be met by ex- pressing V2 in the unprimed coordinate system as

P, cos 4 v, = ~ p:(cos e) [(:)'+I (s - i ) ! 4TU2 s = I

+ C,) * e f l = 1 u n + l ( n - l ) ! ( s - n)!

(-1)"-'(n + l)!a"B, ( s + l ) ! t l = s (n - s)!

After applying appropriate boundary conditions, the following equation for the B,'s is obtained

I illlllllllllll I (I I 11 Ill I 1111 lllllllll Ill ((II/( 1111111

CUFFIN. MODELS OF THE HEAD 875

where same shifts. The shifts produce only small changes in the angular location of the peaks, i .e. , the angle decreases by 3.5” for a positive shift and increases by only 1 So for a negative shift. The peak magnetic fields change by less than 2 % for these shifts and the locations of the peaks are nearly unchanged.

c7\ = k3[sR2‘+I + (s + l)d2\+1] C = R2\f I - d2\+I

SF

These Bn’s are used in the following equation for the potentials on the surface of the model:

P, cos 4 v4 = ~ c

4ao4 PE (COS e) (2s + 1 ) 3 ~ w + Id2\ + IS!

\ = I s2{(sk2 + s + l>d2‘+’[C7, + (s + l)C,,] - (s + 1)(1 - k2)C2\+I[C7\ - SC,,]}

(nk, + n + l)f2”+l + n( l - kl)b2’1+l(f’i+2B,, + 1) n = l (nk , + n + l)fr’+2a‘i+l(n - l)!(s - n ) !

The Be( and B4, components from the surface of the eccentric bubble are

p(1 - k l ) P , sin 4 OD P:, (cos r3’)~2’’+I(f’r+2B,, + I ) BO, - c

4 a sin e’ , , = I (nk , + n + l ) r r r i+1 f r l+2

For a P, dipole, the B,,’s are obtained from

(1 - kl)b’”+’B,, 5 n = I (nk, + n + l )a”+’(n - l)!(s - n)!

(sk2 + s + l)d2‘+l[C7\ + (s + l)CSJ - (s + 1)(1 - k2)c2\+’(C7, - sCXJ - [(s + l ) k * + S]C2\+I(C7\ - SC,,) - s(1 - k2)d2F+1[C7c + (s + l)CS\]

s(s + 1) (y ; a ’I ( - 1 )‘I - ‘n ! B,, X

[(s + l)!]? a n = s (n - s)! ’ (nk, + n + 1)f2”” - (n + 1) (1 - k,)b2”+I

f ’ l + 2 ( n k l + n + l ) a ” + ’ ( n - l)!(s - n)! = - c

t 1 = I

and the potential on the surface of the model is given by

v - L C P ; ( ~ ~ ~ O ) p s + 4nu4 T = i s { ( sk2 + s + l )d* ’+’ [~ , , + (s + l ) ~ ~ , ] - (s + 1)(1 - k 2 ) ~ 2 \ + 1 ( ~ 7 \ - SC,,))

x c

4 -

’ (nkl + II + l)f2”+’ + (1 - k1)b2”+1[f‘i’2B,, - (n + l)] (nk, + n + l)j’i+*a’i+’(n - l)!(s - n) ! r i = I

(43)

(44)

Again, no magnetic field is produced by this dipole.

RESULTS

The potentials and fields produced by dipolar sources in the three models are given in Figs. 2-4. For Model I in Fig. 2, the peak potential for a P; dipole increases by 22 %, as compared with a concentric spheres model with the same parameters, for a 0.25 cm shift of the inner sphere in the positive Z-direction; this shift has the effect of decreasing skull thickness in the region of the peak. A 12% decrease in the peak potential occurs for a 0.25 cm shift in the negative Z-directions; this shift increases skull thickness in the region of the peak. These shifts have no significant effects on the spatial pattern of the potentials, i.e., the general shapes of the curves are the same, only the amplitudes are changed. For a P, dipole, the peak potential increases by 20% and decreases by 12% for the

For Model I1 in Fig. 3, the effects of the shifts on the peak potentials for a Pz dipole are larger than for Model I; the changes are +30% and - 14% for a +0.25 cm and -0.25 cm shift, respectively. A positive shift makes the scalp thinner in the region of the peak and a negative shift makes it thicker. These shifts produce small changes in the spatial pattern of the potentials; at 0 = 40” the curves cross. The changes in the peak potentials for a P, dipole are larger than for Model I, i.e., +31% and -16%. The changes in the angular locations of the peak potentials have approximately the same amplitude and are in the same directions as for Model I. There are some small effects on the spatial patterns; for 8 = 100” the curves cross. The changes in the peak magnetic fields are smaller than for Model I , i.e., less than 1 %. As for Model I, there are nearly no changes in the locations of the peaks.

876 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 9. SEPTEMBER 1991

1.5r

C

Fig. 2. Electric potentials and magnetic fields for Model 1 with b = 8.4 cm. c = 9 .0cm, R = 9.5 cm, and U , = u3 = 800~. The potentials produced by P , and P; dipoles are given in the upper plot. The solid lines are potentials from a concentric spheres model with the same radii and conductivities and f = 6.4 cm. The short-dash lines are potentials from the eccentric sphcres model with a = 0.25 cm andf = 6.15 cm while the long-dash lines are for a = -0.25 cm andf = 6.65 cm: these sets of parameters place the dipoles at the same depth as in the concentric spheres model. The potentials are with respect to a reference on the bottom of the models (0 = 180"). The radial field components for a P , dipole are given in the lower plot; these are the fields at a distance of I cm from the surface of the models.

For Model I11 in Fig. 4, there is an increase in the peak potential for a P, of 11% when the conductivity of the bubble is three times that of the surrounding region and a decrease of 14% when the conductivity is 0.01 times. A conductivity of 3 times is intended to represent a chamber filled with cerebrospinal fluid, e.g. , a ventricle, while a conductivity of 0.01 times is intended to represent a cal- cified tumor. These conductivity changes have no significant effects on the general spatial pattern of the potentials. The pattern of the effects of the changes is reversed for a P, dipole, i.e., there is a 7% decrease for the greater conductivity and an 8 % increase for the smaller conductivity. The angular location of the peak decreases by 2" for the greater conductivity and increases by 1" for the smaller. The peak magnetic field decreases by 5 % for the greater conductivity and increases by 8 % for the smaller. There are no significant changes in the angular locations of the peak fields with changes in conductivity.

Moving dipole inverse solutions using data from the three models are presented in Tables I and 11. In these solutions, a dipolar source is moved about in a model of the head while its amplitude and orientation are also

1lllll1Illl1l1lllll llllll

Fig. 3 . Electric potentials and magnetic fields for Model 11. Model parameters and plots are as in Fig. 2 .

1.5, I

Fig. 4 . Electric potentials and magnetic fields for Model 111. Model parameters and plots are as in Fig. 2 except a = 3.0 cm,f = 3.4 cm, and U?

= u4 = 800~. The potentials and fields for the short-dash lines are for U ,

= 3.00~ and the long-dash lines for U , = 0.010~. The solid lines are for a concentric spheres model with no bubble.

llul1llllllll Ill I II Ill I I I I II I I I Ill

877 CUFFIN: MODELS OF THE HEAD

TABLE I

Model U V I Depth p;

Act. 3. I O 1 .oo I 0.25 - 2.73 1.13 I -0.25 - 3.34 0.92 11 0.25 - 2.31 1 . 1 1 I1 -0.25 - 3.69 0.95 111 - 3.0 3.34 1.16 I11 - 0.01 2.74 0.80

A

-0.37 0.24

0.59 0.24

-0.36

-0.79

Inverse solutions for P; dipoles. The solutions are calculated in the concentric spheres model using data from the eccentric spheres models. Only electric potential solutions are given since the magnetic fields are zero for this dipole. The measurement grid uses 64 points on an 8 x 8 grid as described in Fig. ](a). The potentials use a reference on the bottom of the model. A is the distance between the solution and the actual source. The actual source depth and amplitude are given in the top line of the table. All distances are in centimeters.

TABLE 11 ~~

Model a U , D Depth P , A

Act. 3.10 1.00

I 0.25 - E 2.75 1.12 -0.35 1 0.25 - M 3.07 0.97 -0.03

I -0.25 - E 3.31 0.92 0.2 I

as the potential solutions, but they are much smaller, i.e., the maximum error is 0.06 cm. The maximum error in the solution amplitudes is 13 % .

As part of the inverse solution calculations, a measure of the closeness of fit ( F ) between the data from an eccentric spheres model and that produced by the inverse solution dipole in the concentric spheres model is calculated. This measure is defined as

c (C; - Si)* i = l

F = :" x 100% (45)

where Ci is the eccentric spheres model data at the ith grid point, S, is the moving dipole solution data, and the summation is over the 64 measurement points. The value of F would be zero if the C, data were from a concentric spheres model. All solutions in Tables I and I1 had F values of less than 3 % .

DISCUSSION I -0.25 - M 3.13 1.03 0.03 The equations presented in this paper provide a rapid

E 2.46 -:::: means to calculate the electric potentials and the radial component of the magnetic fields produced by dipolar

I1 -0.25 - M 3.11 1.01 o ,ol sources in three eccentric spheres models of the head. For - 3,0 E 2.96 0.91 -o,14 the particular eccentricities and the conductivities inves-

111 - 3.0 M 3.05 0.95 -0.05 tigated here, the maximum number of terms (smaX) re-

11 0.25 - I1 0.25 - M 3.10 0.97

I1 -0.25 - E 3.51 0.91 0.41

111

111 - 0.01 E 3.25 1.12 0.15 I11 - 0.01 M 3.16 1.06 0.06

Inverse solutions for P , dipoles. All parameters are as for Table I except that the D column indicates an electric ( E ) or magnetic ( M ) data solution. The magnetic solutions use measurements of the radial field component at a distance of I cm from the surface of the model.

changed to obtain the best fit between the data from the eccentric spheres model and that produced by the dipolar source. The inverse solutions are calculated in a concentric spheres model [8] having the same radii and conductivities as Models I and 11. The 8 x 8 measurement grid used is large enough and has enough points on it so that the inverse solutions are not sensitive to small changes in its size or the number and density of measurement points [9]. In Table I, the maximum localization error (A) is 0.79 cm which occurs for Model 11. For Models I and 11, the direction of the error is in the same direction as the shift of the spherical surface. For Model 111, the solution is deeper when the conductivity of the bubble is greater than the surrounding region and more shallow when the conductivity is smaller. The maximum error in solution amplitudes is 20%. The maximum localization error in Table I1 is 0.64 cm which occurs for the potential solution for Model 11. For Models I and 11, the directions of the errors are the same as for the P , dipole in Table I. However, the directions are opposite for Model 111. The directions of the errors for the field solutions follow the same pattern

quired for the calculations was 25. These calculations were performed in a small fraction of the time that would have been required for numerical computer models.

A check of these equations was performed by compar- ing the values obtained from them with values from numerical computer models. For Models I and 11, the computer models contained the same three regions as in Fig. l(a) and (b). However, for Model 111, the computer model contained only the bubble in the brain region. The two concentric outer layers were not included in the computer model and the conductivities of these two regions were set equal to that of the brain region in the equation. The maximum differences between the values from the computer models and the equation was 6% for the potentials and 3% for the fields.

The results obtained using these equations are in agreement with intuition and previous work. The potentials increase when the skull or scalp are made thinner and the reverse occurs for a thicker skull or scalp. The variations in the skull and scalp layer thicknesses and bubble conductivity investigated here had at most only small effects on the general spatial patterns of the potentials or fields. However, some significant effects on the amplitudes were found. The effects on the field amplitudes are significantly smaller than on the potential amplitudes for Models I and 11; the maximum effect on the fields is only 17% of that on the potentials. This is in agreement with results from the previous study 123 using numerical computer models.

878 IEEE TRANSACTIONS ON BlOMEDlCAL ENGINEERING. VOL. 38. NO. 9. SEPTEMBER 1991

The effects on the fields produced by the bubble in the brain region (Model 111) are only somewhat smaller than on the potentials. The effects on the potentials are as pre- dicted by the “Brody” effect [ lo], i.e., when the conductivity of the bubble is greater than the surrounding region, the potential is increased for a dipole perpendicular to the bubble and decreased for a tangential one. There is also a “Brody”-like effect on the fields, i.e., the field decreases for increased conductivity of the bubble. This is in agreement with the results obtained in a previous study [ 111 of the effects of a bubble in a medium of infinite extent.

The largest localization error caused by the eccentricities and conductivities investigated here is 0.79 cm. This occurs for the potential solution for a radial source with variation in the scalp thickness (Model 11). The errors for the field solutions are much smaller, i.e., less than 10% of the maximum potential solution error. While amplitude errors of as much as 20% occur for the eccentricities and conductivities investigated here, this is not a serious con- cem since, for most purposes, it is accurate localization of a source that is important.

The magnitude of the localization errors is closely re- lated to the effects of the model features on the spatial patterns. For example, the largest errors for the potential solutions occur for variations in scalp thickness for which the deviations of the spatial pattems from those of a concentric spheres model are also the largest. While such deviations cause localization errors, the results obtained here indicate that the presence of the deviations and the associated localization errors will not be detectable by the size of the F value of the inverse solution. The F values for the eccentricities and conductivities investigated here are less than 3 % which is much smaller than the F values likely to be caused by noise and measurement location errors [9].

It should be noted that the potential results in Figs. 2-4 are for a reference point on the bottom of the models. If a different reference location were used, the curves would have different shapes. However, the relative differences between the eccentric and concentric spheres potentials would remain the same. Changing the location of the reference would have no effect on the localization errors as long as the inverse solution calculations are also performed using the new reference location.

The equations presented in this paper provide a means to evaluate the effects of variations in skull and scalp

thickness and the presence of a bubble in the brain on EEG’s and MEG’s and source localization accuracy. Using these equations, these effects can be rapidly evaluated over wide ranges of thicknesses, conductivities, etc. using much less computer resources than if numerical computer models were used. The results obtained in this paper using these equations suggest that variations in skull and scalp thickness or the presence of a bubble in the brain will cause localization errors of less than 1 cm for inverse solutions using EEG’s and much smaller errors for solutions using MEG’s.

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B. N. Cuffin, “Eft’ects of head shape on EEG‘s and MEG’s.” IEEE Truris. Biorried. Erig.. vol. BME-37. pp. 44-52. 1990. -, ”A comparison of moving dipole inverse solutions using EEG’s and MEG’s.” IEEE Trms. Biorned. Erig . , vol. BME-32. pp. 905- 910, 1985. D. A. Brody, “A theoretical analysis of intercavitary blood mass in- fluence on the heart-lead relationship,” Cirt.. Res. , vol. 4. pp. 731- 738, 1956. B. N. Cuffin, ”Effects of inhomogeneous regions on electric potentials and magnetic fields: Two special cases.” J . Appl . Phxs.. vol. 53, pp. 9192-9197. 1982.

vol. BME-34, pp. 913-920, 1987.

24, pp. 372-381, 1977.

B. Neil Cuffin received the B.S., M.S. , and Ph.D. degrees in electrical engineering from The Pennsylvania State University, University Park.

He is currently a Staff Scientist at the Francis Bitter National Magnet Laboratory at Massachusetts Institute of Technology, Cambridge. His re- search interests are in studies of the information that measurements of the electrical potentials and magnetic fields of the human body can provide about electrical sources in various organs i n the body such as the heart and brain.

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Eccentric spheres models of the head