Biomedical Engineering Reference
In-Depth Information
As for the case of SPECT, the gamma camera rotates around the patient obtaining
a series of projections taken at different angles which are then combined to form a
tri-dimensional image. Considering now a tomographic acquisition which implies
rotation of the detector, then for a particular projection line it must take into account
the angle,
. Thus, (
8
) generalizes according to:
R
L
0
.
x;y;
z
/
Z
.
x;y;
z
/
ds
0
f.x;y;
z
/
e
p.x;
z
;/ D k
ds:
(9)
L
The problem of determining the radionuclide distribution,
f.x;y;
z
/
, from
the projections,
p.x;
z
;/
, is known as the image reconstruction problem. In
scintigraphy, there is not enough data to determine the radionuclide distribution.
Consequently, the image reconstruction refers only to tomography.
Note that for a given value
z
, the set of projections
p.x;/
is the (attenu-
ated) Radon transform of the object. However, it is common in certain clinical
applications, to assume that attenuation is constant (usually zero), and (
9
) takes a
simpler form:
p.x;
z
;/ D k
Z
f.x;y;
z
/ds;
(10)
L
which constitutes the Radon integral.
Despite (
10
) is only a crude approximation of the measurement process involved
in emission imaging, it is widely applied in SPECT reconstruction methods used in
clinical routine. The images obtained are not quantitatively exact, but qualitatively
they are good enough to permit accurate diagnosis. The problem of quantification in
SPECT is complex since it implies that the attenuation map have to be considered
in the reconstruction step. The method for correcting the attenuation most used in
clinic, because it is easy to apply, was proposed by Chang [
11
] and consists on the
correction of the estimates of the radionuclide distribution slice by slice, according
to the equation:
f
corrected
.x; y/j
z
D
k.x;y/ f
estimate
.x; y/j
z
:
(11)
The map of the weighting,
k.x;y/
, is the inverse of the average attenuation
considered in all the projection lines that contain the point
.x; y/
.Othermore
accurate methods such as analytical and iterative techniques produce better results
but are less used in clinical routine since they are slower. Analytical methods are
generally based on the analytic inversion of the attenuated Radon transform [
12
-
14
].
On the other hand, iterative methods allow to include the attenuation correction into
the model itself.
Regarding the image reconstruction methods, it is usual to distinguish between
analytical and iterative methods.
