Biomedical Engineering Reference
In-Depth Information
and C a ( t ) have been corrected for delay and dispersion. The term V B C a ( t )in
Eq. (2.22) represents intravascular activity present in the FOV of the scanner
and is particularly important at the times immediately following tracer admin-
istration. Note that although C T ( t ) has an analytical representation, it is a non-
linear function of parameters α j and β j , and therefore nonlinear parameter
estimation approach such as the nonlinear least-squares (NLLS) method using
Gauss-Newton type algorithms [64] is required to estimate the parameters α j
and β j , unless the model equation is “linearized” by certain transformations. Al-
ternatively, the rate constant parameters in the system of differential equations
can be estimated directly by numerical method.
Ideally, a comprehensive model describing the full kinetics of a tracer is de-
sirable, but this may require a complicated compartmental model configuration
with many parameters and compartments. Given that the counting statistics of
the acquired data and the spatial resolution of the imaging device are limited, the
compartments and parameters in the complicated compartmental model cannot
be resolved and identified from the measured data. Thus, the actual compartmen-
tal model configuration must be simplified. Typically, measured data from PET
and SPECT can support compartmental models with six parameters or less. In
general, including more number of compartments (or parameters) or increasing
the complexity of the model improves the fit to the measured data. However, the
improvement in the fitting may not be statistically significant. In many cases, the
reliability of individual parameter estimates degrades with increasing number
of compartments (or model complexity). Therefore, the model with the smallest
number of compartments which can fully describe the measured data should be
used.
2.14.3 Input Function
For compartmental models used in PET, one of the compartments represents
blood pool or extravascular space. This compartment can be seen as the input
stage to the model because the tracer is delivered into the tissue through the
blood, as indicated by Eq. (2.22), and therefore the time-activity concentration
for blood is called input function . In other words, the time-activity concentra-
tion of the tracer in a particular tissue is dependent both on the amount of the
tracer delivered to the tissue via the blood supply (the input function) and on the
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