Biomedical Engineering Reference
In-Depth Information
The scalp electric potential difference at a given electrode receives contributions,
in an additive manner, from all voxels. The equation relating scalp potentials and
current density can be conveniently expressed in vector/matrix notation as:
Φ=
KJ
+
c
1
(5.9)
R
N
E
×1
contains the instantaneous scalp electric potential differ-
ences measured at
N
E
electrodes with respect to a single common reference electrode
(e.g., the reference can be linked earlobes, the toe, or one of the electrodes included
in
where the vector
Φ∈
R
NN
×
(
3
)
Φ
); the matrix
K
∈
is the lead field matrix corresponding to
N
V
voxels;
J
E
V
N
V
×
is the current density;
c
is a scalar accounting for the physics nature of
electric potentials, which are determined up to an arbitrary constant; and
1
denotes
a vector of ones, in this case
1
R
(
3
)
1
∈
R
N
E
×1
. Typically
N
E
<<
N
V
, and
N
E
≥
∈
19. In (5.9), the
structure of the lead field matrix
K
is
⎛
T
T
T
⎞
k
k
k
⎜
⎜
⎜
⎜
11
12
1
N
⎟
⎟
⎟
⎟
V
T
T
T
k
k
k
K
=
21
22
2
N
(5.10)
V
T
T
T
k
k
k
⎝
⎠
N
1
N
2
NN
E
E
E
V
1, ...,
N
V
) corresponds to the scalp
potentials at the
e
th electrode due to three orthogonal unit strength dipoles at voxel
v
, each one oriented along the coordinate axes
x
,
y
, and
z
. Equations (5.4) and (5.5)
are examples of the lead field that can be written in closed form, although they cor-
respond to head models that are too unrealistic.
Note that
K
can also be conveniently written as
where
k
ev
∈
R
3×
(for
e
=
1, ...,
N
E
and for
ν =
(
)
KKKK
=
,
,
,
,
K
(5.11)
1
2
3
N
V
R
N
E
×3
where
K
∈
(for
ν =
1, ...,
N
V
) is defined as follows:
⎛
k
k
T
⎞
⎜
⎜
⎜
⎜
1
v
⎟
⎟
⎟
⎟
T
2
v
K
=
(5.12)
v
k
T
⎝
⎠
Nv
E
In (5.9),
J
is structured as
j
j
⎛
⎞
1
⎜
⎜
⎜
⎜
⎟
⎟
⎟
⎟
2
N
V
J
=
(5.13)
j
⎝
⎠
where
j
v
∈
R
3×
denotes the current density at the
v
th voxel, as in (5.2).
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