Biomedical Engineering Reference
In-Depth Information
map of the object, the general equation for the attenuated projection data can
be described by the attenuated Radon transform
∞
f
(
x
,
y
) exp
l
(
x
,
y
)
µ
(
x
,
y
)
ds
dl
r
,θ
p
m
(
r
,θ
)
=
(2.18)
−
0
0
where
p
m
(
r
,θ
) is the measured projection data,
l
(
x
,
y
) is the distance from the
detector to a point (
x
,
y
) in the object, while
l
r
,θ
and
r
have the same definitions
as in equations (2.6) and (2.7). It should be noted that unlike the unattenuated
Radon transform as in equation (2.6), there is no analytical inversion formula
available for equation (2.18).
The attenuation correction in PET is simpler and easier as compared to
SPECT due to the difference in the photon detection schemes. In SPECT, the
attenuation depends not only on the source position, but also on the total path
length that the photon travels through the object. It is not straightforward to
correct for attenuation or find an inversion of equation (2.18) for image recon-
struction. On the contrary, the attenuation in PET is independent of the source
position because both gamma rays must escape from the body for external
detection and the LoR can be determined. Therefore, the exponential term in
equation (2.18) can be separated from the outer integral. The unattenuated pro-
jection data and the measured projection data can then be related as follows:
p
m
(
r
,θ
)
=
p
(
r
,θ
)
p
µ
(
r
,θ
)
(2.19)
where
p
(
r
,θ
) is the unattenuated projection data, and
p
µ
(
r
,θ
)
=
exp
∞
µ
(
x
,
y
)
dl
r
,θ
(2.20)
−
0
is the projection data of the attenuation map. Therefore, if the attenuation coeffi-
cient map
µ
(
x
,
y
) or its projection data
p
µ
(
r
,θ
) is known, then the unattenuated
projection data
p
(
r
,θ
) of the object can be calculated as:
p
m
(
r
,θ
)
p
µ
(
r
,θ
)
p
(
r
,θ
)
=
(2.21)
and
f
(
x
,
y
) can then be reconstructed without attenuation artifacts.
Since the attenuation is always the same regardless of the source position
inside the FOV, it is possible to use an external (transmission) positron-emitting
source that comprises a fixed ring or rotating rod sources, to measure the attenu-
ation correction factors through two extra scans:
blank scan
and
transmission
scan
. A blank scan is acquired with nothing inside the FOV, and a transmission
