Biomedical Engineering Reference
In-Depth Information
Lorentz force, oriented along the
x
axis and proportional to
the current and the magnetic field, induces a displacement
l
max
of the conductor in the
x
direction, leading to a spatially
incoherent displacement
l(
x
,
z
) of the surrounding elastic
medium in the
x
direction. We assume that the deformation
is elastic, i.e., the displacement is proportional to the applied
force and inversely proportional to the Young's modulus of the
elastic medium (Hooke's law). Furthermore, we assume that the
conductor does not adhere to the elastic medium, so that it only
induces a compression of the elastic medium on one side, but
no dilation on the opposite side, thus leaving an empty space
behind it. Because of the symmetry with respect to the
z
axis, we
consider only the
region from now on. The maximum
displacement experienced by the elastic medium is equal to the
displacement of the conductor
{
z
≥
0
}
l
max
and occurs at (
x
=
l
max
,
z
0). As a first order approximation, we assume that:
1.
(1)
There is no displacement in the
=
{
x
≤
0
}
,
{
x
≥
L
}
,and
{
z
≥
regions (where
L
and
M
are defined in
Fig. 14.2
);
2. There is an empty space in the
M
}
{
<
<
}
0
x
l
max
(
M
-
z
)/
M
region (hatched in
Fig. 14.2
); (3) The displacement
l
at an
arbitrary point (
x
,
z
)inthe
{
l
max
(
M
-
z
)/
M
≤
x
≤
L
}
region
decreases linearly from
l
max
to zero as follows:
L
−
x
M
−
z
l
(
x
,
z
)
=
l
max
(14.1)
l
max
M
−
z
M
L
−
M
In this region, the spin density is therefore equal to
L
ρ
(
z
)
=
ρ
,
(14.2)
l
max
M
−
z
L
−
M
where
is the spin density in the absence of Lorentz force, and
the phase shift obtained by applying
n
cycles of oscillating gra-
dients (whose positive lobes are synchronized with the current)
along the
x
axis is given by
ρ
n
T
φ
(
x
,
z
)
=
γ
G
l
(
x
,
z
) d
t
≈
γ
nGT
l
(
x
,
z
),
(14.3)
0
where
G
and
T
are the amplitude and duration of one gradient
lobe respectively, and
×
10
6
rad/T for protons). This phase shift is thus directly propor-
tional to the displacement. To derive the right-hand side of
Equa-
tion (14.3)
, the displacement was assumed to occur over a time
much shorter than
T
. If this were not the case, the resulting phase
shift would be smaller.
The ratio of the signal intensity with and without Lorentz
effect in a voxel of dimensions
L
γ
is the gyromagnetic ratio (2
π
×
42. 57
×
M
can be computed by inte-
grating the phase shift over the
{
0
≤
x
≤
L
;0
≤
z
≤
M
}
region:
