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 (magnetoencephalograms) 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.
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 comparing the solutions with the actual known sources.
INTRODUCTION PHERICAL MODELS of the head are often used in analyzing electroencephalograms (EEG’s) and magnetoencephalograms (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 usually 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.
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]
S
Manuscript received July 12, 1990; revised October 25. 1990. This work was supported by NIH Grant NS22703. The author is with the Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge. MA 02 139. IEEE Log Number 9101997.
DERIVATION OF EQUATIONS
4 v, = PI4cos ~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 s+
I
P; (COSe) (s
- I)!
s=n
+ Dnun
s= 1
(:I
+
(-i)”-yn I ) ! P (COS ~ e) ( n - s)!(l + s)!
and after interchanging the order of summation
0018-9294/91/0900-0871$01 .OO 0 1991 IEEE
]
(4)
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IEEE TRANSACTIONS ON BlOMEDlCAL ENGINEERING, VOL 38. NO. 9. SEPTEMBER 1991
Equations (6)-(10) can be solved to obtain the following equation for the Dn's:
k1)b2"fID,7 5 (nkl + n n+( l l)a"+'(n l)!(~ -
-
n = ~
-
n)!
5 u n ( - l ) f l - F ( n+ l>!D" (n - s)!
n=T
=-E S
(nk,
r7=l
(2n + l ) f " - ' + n + l)a"+'(n -
l)!(s
-
n)! (1 1)
where
C , , = k2[sR2'+'+ ( s C = R2S t I - c 2 S + 1
+ l)c2'+l]
2s
and k l = u I / u 2 and k2 = u2/u3. These D,'s can then be used in the following equation for the potential on the surface of the model: cos + v, = P,4nu3 ~
n=l
(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 I1 is shown in (b) and Model 111 in (c).
P, cos
+ 5 Pi (cos e) [
v 2 = _____ 4TU2 s=l
(S
+ l)!
n=s
+ 1)2RSaS+ Is!P,' (cos e) c (2s s2[C,, + (s + l)C2J
s=1
(2n (nkl
+ l)f"-' + n ( l - k1)b2"+'D, + n + l ) a " + ' ( n - l ) ! ( ~- n)!.
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 of v,. The equations for the magnetic field B produced by a P, dipole are derived using the vector magnetic potential A from which
B = V x A .
(s - l)!
(;J+I
(12)
(13)
The equation for A is [6]
(n - s)!
The constants B,, C,,, etc. can be evaluated by applying the boundary conditions VI = V 2 , r' = b
(6) (7)
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 ~ (n + m)!
e,)
(15)
lI l lIl1l l l l l lll
lul1lllIl
I I Il III I I I Il
CUFFIN: MODELS OF THE HEAD
873
where E , = 1 for m = 0 and E , = 2 for m # 0. The coordinates with the s-subscript are at the source or on the spherical surfaces. For this model, it is only necessary to evaluate the surface integral in the second term in (14) over the surface of the shifted sphere. This is because the surfaces of the unshifted spheres produce only Bo and B, components [7]. Hence, it is only necessary to derive equations for the Bo. and B,, components produced by the shifted sphere and then calculate their projections in the i direction to obtain the radial field component. Note that no B, or B,, components are produced by the spherical surfaces [7]. The electric potential produced on the shifted surface is given by
5
cos 4 v, = P, -
(2n
+
l)P;,(cOs
c (nkl =
- s(s
+ 1)
n=s
(3
[CIS
,sf!
[($ + 1YI2
S
(nkl
v 3 = -P,c
(2n + 1)fn-I + n + l ) a n + ' ( n - l)!(s
(2s
4au3 s =
-
(23)
n)!
+ i ) 2 ~ . y a s + -+ ' (i)!p:(cos ~ e) [Cl, + (s
I
+
1)C2Sl
(2n + I ) f " - ' + (1 - k l ) b 2 " f 1 D n . e' (kin + + I ) u " + ' (-~ I)!(s - n ) ! '
n=l
12
n=l
geometry, a
Because Of the duces no magnetic fie1d.
pz
(24)
dipo1e Pro-
Model Inner Two Surfaces shifted The general expressions for the potentials produced by a P, dipole in regions 1 and 2 of the model shown in Fig.
~
P,i x L 4aL3
1)Cd
and the potential on the surface of the model is given by
4TU, [n(kl + 1 ) + 1]6"+' - U - '+ bZn+lD,)
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
+ (s +
(CIS - sC2.J
a"(-l)"pSn!D, (n - s)!
= - nE= l
e l )
(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 evaluated integral, the following equations for the field produced by the shifted surface are obtained:
(1 - kl)bzn+lDIl ~ + l)an + I ( - n)!
+
~
l(b) are the same as ( 1 ) and (2). In region 3, primed coordinates are also used
Bd=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
In order to meet the boundary condition at r is expressed in unprimed coordinates as
=
R , (25)
S
* c~ " + ' (-nEnI)!(s - n>! n = ~
(S
+ l)!
c (-1)"-"n(n
n=s
+ -
l)!a"F, s)!
After applying appropriate boundary conditions, the following equation for the F,I's is obtained: S nC2n + 1 (C3n + C d F n fl=I (csnf C6n)anf1(n - I)!($ - n)!
c
V3
c
pz Ps (cos 0) 4au3 s = I
= -
S2
-
[(s
+ 1)!]2
i:) 2 2s+l
m
an(-l)"-s((n
+ l)!Flf
(n - s)!
+ 1)2fn-1C2n+l - e (c5n+(2nC6n)anf1(n - l)!(s S
The equation from which the D,'s are calculated for a P, dipole is
=
-
n)!
(27)
874
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 38, NO. 9, SEPTEMBER 1991
where
and the surface potential is given by
( 1 - kl)[(n + l)k2
C3, =
c4,
= (1
- kJ(nk1
+ n]b2"+I
+n +
v3
1)c2'1+1
=
5 4nu3 5I (;)
2s
+I (2s
s=
+
l ) ( s - l)!P:(CoS e)u"l
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
_v3_ = _ P,cosu5 : . 4nu3
xc
+
J
I
(E)
,f.I
(2n
S
Again, the magnetic field is zero for this dipole.
+ l~s!P.!(cos8\ -,
(2s
\
~
Model III: Eccentric SDhericul Bubble in the Bruin
S2
1)2fn-I
2n+1
+
,,2r1+1
(eK.. + C
87 1
Eccentric Spheres Models of the Head B. Neil Cuffin
Abstract-Equations are derived for electric potentials (electroencephalograms) and magnetic fields (magnetoencephalograms) 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.
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 comparing the solutions with the actual known sources.
INTRODUCTION PHERICAL MODELS of the head are often used in analyzing electroencephalograms (EEG’s) and magnetoencephalograms (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 usually 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.
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]
S
Manuscript received July 12, 1990; revised October 25. 1990. This work was supported by NIH Grant NS22703. The author is with the Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge. MA 02 139. IEEE Log Number 9101997.
DERIVATION OF EQUATIONS
4 v, = PI4cos ~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 s+
I
P; (COSe) (s
- I)!
s=n
+ Dnun
s= 1
(:I
+
(-i)”-yn I ) ! P (COS ~ e) ( n - s)!(l + s)!
and after interchanging the order of summation
0018-9294/91/0900-0871$01 .OO 0 1991 IEEE
]
(4)
a12
IEEE TRANSACTIONS ON BlOMEDlCAL ENGINEERING, VOL 38. NO. 9. SEPTEMBER 1991
Equations (6)-(10) can be solved to obtain the following equation for the Dn's:
k1)b2"fID,7 5 (nkl + n n+( l l)a"+'(n l)!(~ -
-
n = ~
-
n)!
5 u n ( - l ) f l - F ( n+ l>!D" (n - s)!
n=T
=-E S
(nk,
r7=l
(2n + l ) f " - ' + n + l)a"+'(n -
l)!(s
-
n)! (1 1)
where
C , , = k2[sR2'+'+ ( s C = R2S t I - c 2 S + 1
+ l)c2'+l]
2s
and k l = u I / u 2 and k2 = u2/u3. These D,'s can then be used in the following equation for the potential on the surface of the model: cos + v, = P,4nu3 ~
n=l
(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 I1 is shown in (b) and Model 111 in (c).
P, cos
+ 5 Pi (cos e) [
v 2 = _____ 4TU2 s=l
(S
+ l)!
n=s
+ 1)2RSaS+ Is!P,' (cos e) c (2s s2[C,, + (s + l)C2J
s=1
(2n (nkl
+ l)f"-' + n ( l - k1)b2"+'D, + n + l ) a " + ' ( n - l ) ! ( ~- n)!.
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 of v,. The equations for the magnetic field B produced by a P, dipole are derived using the vector magnetic potential A from which
B = V x A .
(s - l)!
(;J+I
(12)
(13)
The equation for A is [6]
(n - s)!
The constants B,, C,,, etc. can be evaluated by applying the boundary conditions VI = V 2 , r' = b
(6) (7)
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 ~ (n + m)!
e,)
(15)
lI l lIl1l l l l l lll
lul1lllIl
I I Il III I I I Il
CUFFIN: MODELS OF THE HEAD
873
where E , = 1 for m = 0 and E , = 2 for m # 0. The coordinates with the s-subscript are at the source or on the spherical surfaces. For this model, it is only necessary to evaluate the surface integral in the second term in (14) over the surface of the shifted sphere. This is because the surfaces of the unshifted spheres produce only Bo and B, components [7]. Hence, it is only necessary to derive equations for the Bo. and B,, components produced by the shifted sphere and then calculate their projections in the i direction to obtain the radial field component. Note that no B, or B,, components are produced by the spherical surfaces [7]. The electric potential produced on the shifted surface is given by
5
cos 4 v, = P, -
(2n
+
l)P;,(cOs
c (nkl =
- s(s
+ 1)
n=s
(3
[CIS
,sf!
[($ + 1YI2
S
(nkl
v 3 = -P,c
(2n + 1)fn-I + n + l ) a n + ' ( n - l)!(s
(2s
4au3 s =
-
(23)
n)!
+ i ) 2 ~ . y a s + -+ ' (i)!p:(cos ~ e) [Cl, + (s
I
+
1)C2Sl
(2n + I ) f " - ' + (1 - k l ) b 2 " f 1 D n . e' (kin + + I ) u " + ' (-~ I)!(s - n ) ! '
n=l
12
n=l
geometry, a
Because Of the duces no magnetic fie1d.
pz
(24)
dipo1e Pro-
Model Inner Two Surfaces shifted The general expressions for the potentials produced by a P, dipole in regions 1 and 2 of the model shown in Fig.
~
P,i x L 4aL3
1)Cd
and the potential on the surface of the model is given by
4TU, [n(kl + 1 ) + 1]6"+' - U - '+ bZn+lD,)
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
+ (s +
(CIS - sC2.J
a"(-l)"pSn!D, (n - s)!
= - nE= l
e l )
(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 evaluated integral, the following equations for the field produced by the shifted surface are obtained:
(1 - kl)bzn+lDIl ~ + l)an + I ( - n)!
+
~
l(b) are the same as ( 1 ) and (2). In region 3, primed coordinates are also used
Bd=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
In order to meet the boundary condition at r is expressed in unprimed coordinates as
=
R , (25)
S
* c~ " + ' (-nEnI)!(s - n>! n = ~
(S
+ l)!
c (-1)"-"n(n
n=s
+ -
l)!a"F, s)!
After applying appropriate boundary conditions, the following equation for the F,I's is obtained: S nC2n + 1 (C3n + C d F n fl=I (csnf C6n)anf1(n - I)!($ - n)!
c
V3
c
pz Ps (cos 0) 4au3 s = I
= -
S2
-
[(s
+ 1)!]2
i:) 2 2s+l
m
an(-l)"-s((n
+ l)!Flf
(n - s)!
+ 1)2fn-1C2n+l - e (c5n+(2nC6n)anf1(n - l)!(s S
The equation from which the D,'s are calculated for a P, dipole is
=
-
n)!
(27)
874
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 38, NO. 9, SEPTEMBER 1991
where
and the surface potential is given by
( 1 - kl)[(n + l)k2
C3, =
c4,
= (1
- kJ(nk1
+ n]b2"+I
+n +
v3
1)c2'1+1
=
5 4nu3 5I (;)
2s
+I (2s
s=
+
l ) ( s - l)!P:(CoS e)u"l
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
_v3_ = _ P,cosu5 : . 4nu3
xc
+
J
I
(E)
,f.I
(2n
S
Again, the magnetic field is zero for this dipole.
+ l~s!P.!(cos8\ -,
(2s
\
~
Model III: Eccentric SDhericul Bubble in the Bruin
S2
1)2fn-I
2n+1
+
,,2r1+1
(eK.. + C
