Biomedical Engineering Reference
In-Depth Information
To introduce the basic form of the equations, consider the nonrealistic, simple
case of the current dipole in an infinite homogenous medium with conductivity
.
R
3×1
for
r
e
≠
Then the electric potential at location
r
e
∈
r
v
is
(
)
φ
rrj
,,
=
k j
T
+
c
(5.3)
e
vv
e v
,
v
where
(
)
r
−
r
1
e
v
k
=
(5.4)
ev
,
4
πσ
3
r
−
r
e
v
denotes what is commonly known as the lead field. In (5.3),
c
is a scalar accounting
for the physics nature of electric potentials, which are determined up to an arbitrary
constant.
A slightly more realistic head model corresponds to a spherical homogeneous
conductor in air. The lead field in this case is
⎡
⎤
(
)
(
)
rr
−+ −
r
r
r r
r
−
r
1
e
v
ee
v
e
v
e
k
=
⎢
⎢
2
+
⎥
⎥
(5.5)
[
]
ev
,
3
(
)
4
πσ
T
rr r rr r
−
−
+
rr r
−
r
−
r
⎣
e
e
v
e
e
v
e
e
v
⎦
e
v
in which this notation is used:
(
) (
)
2
X
=
tr
X
T
X
=
tr
XX
T
(5.6)
and where
tr
denotes the trace, and
X
is any matrix or vector. If
X
is a vector, then
this is the squared Euclidean
L
2
norm; if
X
is a matrix, then this is the squared
Frobenius norm.
The equation for the lead field in a totally realistic head model (taking into
account geometry and full conductivity profile) is not available in closed form, such
as in (5.4) and (5.5). Numerical methods for computing the lead field can be found
in [22].
Nevertheless, in general, the components of the lead field
k
e,v
=
(
k
x
k
y
k
z
)
T
have a very simple interpretation:
k
x
corresponds to the electric potential at position
r
e
, due to unit strength current dipole
j
x
=
1 at position
r
v
; and similarly for the other
two components.
Formally, we are now in a position to state the single-dipole fitting problem. Let
φ
e
(for
e
1, ...,
N
E
) denote the scalp electric potential measurement at electrode
e
,
where
N
E
is the total number of cephalic electrodes. All measurements are made
using the same reference. Let
=
1, ...,
N
E
) denote the theoretical poten-
tial at electrode
e
, due to a current dipole located at
r
v
with moment
j
v
. Then the
problem consists of finding the unknown dipole position
r
v
and moment
j
v
that
best
explain the actual measurements. The simplest way to achieve this is to minimize the
distance
between theoretical and experimental potentials.
Consider the functional:
φ
e
(
r
v
,
j
v
) (for
e
=
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