Biomedical Engineering Reference
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modified to an alternating minimization scheme
B
h
K
T
1
B
h
K
T
A
k
g
KB
h
A
k
A
k+1
=
B
a
K
T
1
B
a
K
T
.
h
k
g
KB
a
h
k
h
k+1
=
Here the notation B
a
and B
h
is indicating whether the coecients or the
input curve in the discretization of Equations (9.5) and (9.6) are kept con-
stant during the particular EM-step. The proposed method is a combined re-
construction method; the parameters that were identified are an input curve
and coecients with respect to an exponential basis. Figure 9.2 shows H
2
15
O-
reconstructions of a full 4D dataset consisting of 26 3D frames, reconstructed
with the standard EM-algorithm and with the modified 4D-EM-algorithm,
respectively.
9.2.3 4D reconstruction methods incorporating non-linear
parameter identication
Since applications such as myocardial perfusion quantification or regional
cerebral glucose measurement involve the parameter identification of non-
linear parameters, it is natural to combine non-linear parameter fitting and
the PET reconstruction process. One option is to compute 4D EM reconstruc-
tions as described in Section 9.2.2 and to post-process these reconstructions
via a non-linear parameter fitting. The obvious drawback is that the method of
Section 9.2.2 is unable to capture the non-linear behavior of the particular pa-
rameters due to the linearity of the basis operators B. Hence, it seems natural
to incorporate the non-linear model into the reconstruction process to enhance
the parameter identification process. In [3], different variants of Equation (9.4)
have been incorporated into the reconstruction process via forward-backward
splitting. Results obtained from Monte-Carlo simulated synthetic H
2
15
O-PET
data involving parameter identifications for the parameters perfusion, arterial
spillover and an arterial input function located in a left ventricular area can
be seen in Figure 9.3. The data set consists of 26 temporal frames, and the
spatial dimension is two- and not three-dimensional, though the whole proce-
dure could easily be extended to full 4D. Other works on the combination of
kinetic parameter estimation and PET reconstruction can be found, e.g., in
[65, 66, 46, 47, 48].
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