Cinderalla Solution

Linearization approaches reformulate the model equations so that (1) a linear relationship exists between the transformed data and the primary physiological parameter of interest, or (2) the reformulated model equations contain only linear parameters. In these circumstances, estimation of parameters can be accomplished by a simple linear regression or by linear least-squares (LLSs) techniques.

A number of graphical techniques that aim at transforming the measured data into a plot which is linear after a certain "transformed time" have been proposed for specific tracer studies, including the Patlak [75,76], Logan [77, 72], and Yokoi [78, 79] plots. Applications of the techniques depend on the tracer studies and parameter of interest. The Patlak plot [75] was initially developed for estimating the influx rate constant of radiotracer accumulation in an irreversible compartment, and was extended to allow for slow clearance from the irreversible compartment [76]. When employed in FDG studies, the influx rate constant is directly proportional to the regional metabolic rate of glucose. The Logan plot [77, 72] was primarily developed for estimation of parameters related to receptor density such as binding potential and volume of distribution for neuroreceptor studies and the radiotracers can have reversible uptake. The Yokoi plot [78, 79] has been proposed as a rapid algorithm for cerebral blood flow measurements with dynamic SPECT. Although all these methods permit the estimation of physiologic parameter in rapid succession and have been used extensively because of their computational simplicity, the bias introduced into the physiologic parameters is significant in the presence of statistical noise in the image data.

The use of linearized model equations was first proposed by Blomqvist [80] for the Kety-Schmidt one-compartment model used for measuring cerebral blood flow [81] and was extended by Evans [82] for the three-compartment model (as shown in Fig. 2.9) to measure cerebral metabolic rate of glucose. The key idea is that by reformulating and integrating the model equations, the 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 measurement 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].

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