Combined Correction and Reconstruction Methods - Correction Techniques in Emission Tomography - page 189

the tracer distribution in the left ventricular region, h(t), and the myocardial
perfusion, p, can be related to each other via

Z t

h() exp p

(t) d ,

f(t) = p (9.4)

0

with denoting the so-called partition coecient, usually set to a value be-
tween 0.9 and 1. Modifications of (9.4) have been developed, for instance [23].
Not only myocardial perfusion but also other applications involve non-linear
parameter identification, as for example the measurement of regional cerebral
glucose use.
However, the capabilities of compartment models are limited. Due to the
restriction of investigating tracer activities in homogeneous spatial regions
only, important physical effects (like motion) are completely neglected. Hence,
a parameter identification of physiological parameters obtained with a model
derived from compartment modeling could easily be distorted by motion ar-
tifacts. Moreover, in standard compartment modeling exchange between dif-
ferent spatial regions (e.g., voxels) is usually not taken into account, leading
to ODE-type instead of PDE-type models.
For more information on compartment modeling we refer to [64].

9.2.2 4D reconstruction methods incorporating linear pa-
rameter identification
In [50] a method has been proposed that estimates a set of temporal basis
functions and a set of coecients for each voxel simultaneously. Two different
models for the mean of the measured PET data have been used to perform
an alternating minimization.
In order to combine the reconstruction process with compartment mod-
eling, in [49] a method has been proposed that assumes every voxel to be a
Laplace convolution of an abstract input curve and a set of N predefined ex-
ponential basis functions, as described in (9.3). This means, the desired image
sequence is assumed to be of the type

X

a n (x)b n (t) ,

f(x;t) = (9.5)

n=1

with

Z t

b n (t) =

h() exp (b n (t)) d . (9.6)

0

If we discretize the coecients a n (x), the basis functions b n (t) and the input
function h(t) in space and time, respectively, and denote these discretizations
as matrices A and B and a vector h, the standard EM-algorithm can be