Biomedical Engineering Reference
In-Depth Information
operational equations will be linear in the parameters to be estimated, whereby
linear least-squares or weighted linear least-squares methods can be used to
estimate the parameters of interest. While the measurement errors are typically
statistically independent in time, integration introduces correlation of measure-
ment errors, which can introduce bias into the parameter estimates [83]. The
generalized linear least-squares method was designed to remove bias in the
estimates resulting from integration of measurements and has been extended
to multicompartment models and has been found useful in fast generation of
parametric images [84-86].
2.14.6 Spectral Analysis
In compartmental model fitting, the number of compartments and their intercon-
nection are defined
a priori
. This implies that the physiological or biochemical
pathways are somewhat known. Yet,
a priori
knowledge about the behavior of
novel anticancer drugs may not be available. Further, the compartmental mod-
eling approach assumes well-mixed, homogeneous tracer distribution within
the tissue or the ROI. This may not be true for tumor which normally has high
degree of heterogeneity.
Spectral analysis
does not rely on tracer assumptions
and the number of compartments and their connectivity; it is particularly useful
for tracer kinetics studies.
Spectral analysis [87] fits the model defined in equation (2.22) with a prede-
fined set of basis functions,
e
β
j
t
⊗
C
p
(
t
), where
β
j
can take on a discrete set of
values so that a large number (100 or more) of basis functions are generated. The
fitting to tissue data is accomplished by nonnegative least squares (NNLS) algo-
rithm with a constraint
α
i
≥
0 [88]. Typically, a linear combination of only two
or three basis functions from the complete set of basis functions are identified
which can best describe the observed tissue data. From the fitted basis func-
tions, the impulse response function and other physiological parameters can
be estimated. Spectral analysis can also be applied to projection data directly,
but it may not produce results equivalent to those obtained from reconstructed
images because the NNLS fitting may not be linear [89].
Since spectral analysis does not require any
a priori
definition of the nu-
merical identifiable components present in the PET data, it is more flexible than
compartmental model fitting. However, the assumption on the nonnegativity co-
efficients of exponentials may not be valid in a generic compartmental model as
