Biomedical Engineering Reference
In-Depth Information
We then apply the Taylor series expansion:
e
−
ik
|
r
−
r
|
r
|+
...
1
−
ik
|
r
−
(A.35)
r
|
<
For MEG and EEG applications, we assume that
1 m, typical for head-
radius a
nd di
stance from cortical sources to the sensors. With
|
r
−
0
.
ˉ/˃
1, we have
|
√
ˉμ
|
k
. Head tissue conductivity is highly variable, but the nominal value is
r
|≈
˃
≈
0013. Thus, propagation delays
may also be ignored to within a 1% error. With these simplifications, we can obtain
0.1 s/m. Using this value leads to
|
k
||
r
−
0
.
r
)
μ
0
4
J
S
(
d
3
r
,
A
(
r
)
=
(A.36)
ˀ
|
r
−
r
|
V
and
∇
·
r
)
J
S
(
1
d
3
r
.
ʦ
(
)
=−
r
(A.37)
ˀ˃
|
−
r
|
4
r
V
These simplified solutions are termed the quasi-static solutions, because the depen-
dence on
has been eliminated. This rigorous derivations often ignored in MEG
and EEG literature, are given here to make explicit assumptions about tissue para-
meters and source frequencies which underly the quasi-static formulations of EEG
and MEG.
ˉ
A.2.2.3
Computation of Magnetic Field and Electric Potential
From the magnetic vector potential, the magnetic field is computed using:
r
)
μ
0
4
J
S
(
d
3
r
.
B
(
r
)
=∇×
A
(
r
)
=
V
∇×
(A.38)
ˀ
|
−
r
|
r
We use the identity:
r
)
J
S
(
1
1
r
)
+
r
)
∇×
r
|
=∇
r
|
×
J
S
(
r
|
∇×
J
S
(
|
r
−
|
r
−
|
r
−
r
)
×
(
r
)
J
S
(
−
r
=
.
(A.39)
|
−
r
|
3
r
r
|
−
1
r
)/
|
r
|
−
3
Note that since
∇
is applied to
r
, the relationships
∇|
r
−
=−
(
r
−
r
−
r
)
=
and
0 hold. Therefore, themagnetic field in an unbounded homogeneous
medium is expressed as
∇×
J
S
(
r
)
×
(
r
)
μ
0
4
J
S
(
r
−
d
3
r
.
B
(
r
)
=
(A.40)
r
|
3
ˀ
|
r
−
V
