YAN et al.: DIRECT 4-D PET LIST MODE PARAMET RIC RECONSTRUCTION W ITH A NOVEL EM ALGORITHM 2215
Fig. 1. Illustrati on of static and dynamic complete data spaces. (a) Static spatial
complete data
. (b) Plasma inpu t, colors represent different delivery times.
(c) Time activity curve and separation of activity b ased on tracer delivery time.
(d) Spatiotemp oral complete data
. See text for details.
all of the previous input function valu es. The convolution prop-
erty indicates that the later a given molecule of tracer is deliv-
ered to the t issue, the greater is i ts contribution t o the activity
. A simple example is shown in Fig . 1(b), the input func-
tion, and Fig. 1(c), th e tissue TAC. The activi ty at t ime 20 min
has four com ponents, with each component derived from the
plasma input from different times, denoted with the matching
colors. This ability to separat e the activity a t a given time into
its components, based on the arrival time in the plasma, moti-
vates the definition of the complete data space for a 4-D EM
algorithm [as shown i n Fig. 1(d)], w here different colors repre-
sents different arrival times). Thus, we define a spatiotemporal
complete data space
as the counts collected along LOR
in time bin emitted from voxel and where the tracer input that
produced that event was delivered at tim e
.Forthe1Tmodel,
this random variable is
(7)
From the complete data definition, the E- and M-steps of the EM
algorithm for estimation of the parameters can be derived. In a
similar manner to the static EM algorith m, the E-step calculates
the expectation of the com plete data given measured data
,and
the current kinetic parameters estimates
and at iteration
(8)
The M-step maximizes the corresponding conditional complete
data likelihood, which, ignoring constant terms i s
(9)
Taking partial derivatives with respect to
, and setting to
zero, yields
(10)
whichleadstothe
update equation, in terms of
(11)
where
, the voxel sensitivity. F or
(12)
The
update ( 11) is inser ted into (12), wh ich leads to
(13)
The right-hand side of (13) is a fun c tion of the measured data
and the current parameters (8). The left-hand side i s a function
only of
and the known input function; we define this term
as
. This function can be precomputed for a range of
values and is fo und to be mo notoni cally decreasing. Therefore,
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