Biomedical Engineering Reference
In-Depth Information
=
(
,
,
)
A spatial location is represented by a three-dimensional vector
r
:
r
x
y
z
.
(
,
)
A source vector is defined as a three-dimensional column vector
s
r
t
:
⊡
⊣
⊤
⊦
,
s
x
(
,
)
r
t
s
y
(
,
)
s
(
r
,
t
)
=
r
t
(2.2)
s
z
(
r
,
t
)
where
s
x
(
are the
x
,
y
, and
z
components. The physical
nature of the source vector is the electromotive force generated by neuronal activities
in the brain. Additional discussion regarding the nature of the sources is presented
in Sect. A.1 in the Appendix. The magnitude of the source vector is denoted as a
scalar
s
r
,
t
)
,
s
y
(
r
,
t
)
, and
s
z
(
r
,
t
)
(
r
,
t
)
, and the orientation of the source is denoted as a three-dimensional unit
T
, where the superscript
T
indicates the matrix
vector
ʷ
(
r
)
=[
ʷ
x
(
r
), ʷ
y
(
r
), ʷ
z
(
r
)
]
transpose. Then, the relationship
⊡
⊤
ʷ
x
(
r
)
⊣
⊦
s
(
r
,
t
)
=
s
(
r
,
t
)
ʷ
(
r
)
=
s
(
r
,
t
)
ʷ
y
(
r
)
(2.3)
ʷ
z
(
r
)
holds.
2.3 Sensor Lead Field
We assume that a unit-magnitude source exists at
r
. We denote the output of the
m
th
sensor due to this unit-magnitude source as
l
m
(
,
l
m
(
, and
l
z
m
(
when the unit-
magnitude source is directed in the
x
,
y
, and
z
directions, respectively. The column
vectors
l
x
(
r
)
r
)
r
)
r
)
,
l
y
(
r
)
, and
l
z
(
r
)
are defined as
l
1
(
l
2
(
l
x
M
(
T
l
x
(
r
)
=[
r
),
r
),...,
r
)
]
,
l
1
(
l
2
(
l
y
M
(
T
l
y
(
r
)
=[
r
),
r
),...,
r
)
]
,
l
1
(
l
2
(
l
z
M
(
T
l
z
(
r
)
=[
r
),
r
), . . . ,
r
)
]
.
These vectors express the sensor array sensitivity for a source located at
r
and directed
in the
x
,
y
, and
z
directions. Using these column vectors, the sensitivity of the whole
sensor array for a source at
r
is expressed using an
M
×
3 matrix:
L
(
r
)
=[
l
x
(
r
),
l
y
(
r
),
l
z
(
r
)
]
.
(2.4)
This matrix
L
(
r
)
is called the lead-field matrix. We also define the lead-field vector,
l
(
r
)
, that expresses the sensitivity of the sensor array in a particular source direction
ʷ
(
r
)
, such that
l
(
r
)
=
L
(
r
)
ʷ
(
r
).
(2.5)
