Graphical analysis of PET data applied to reversible and irreversible tracers
Introduction
Graphical analysis refers to the transformation of multiple time measurements of plasma and tissue uptake data into a linear plot, the slope of which is related to the number of available tracer binding sites. This type of analysis allows easy comparisons among experiments. No particular model structure is assumed which can be an advantage in many cases since one model may not describe equally well all data sets from the same region of interest (ROI). It is assumed that the plasma concentrations of unchanged tracer are monitored following tracer injection. The requirement of plasma measurements can be eliminated in some cases when a reference region is available. There are two categories of graphical methods that apply to two general types of ligands—those that bind reversibly during the scanning procedure (14) and those that are irreversible or trapped during the time of the scanning procedure 1, 11, 20, 21, 25, 26. This distinction between reversible and irreversible depends on the length of the experiment. A ligand might be reversible over a period of many hours or days, but for the hour or so of the experiment it could be considered irreversible. It is not always possible to distinguish reversible from irreversible tracers from uptake data alone.
Graphical analysis techniques have been applied extensively to tracers used in neuroreceptor imaging, but their application is not limited to these studies. The examples presented here are taken from receptor studies in the brain, but the problems illustrated will be relevant to other applications.
Section snippets
Graphical analysis of reversible ligands
For reversible systems the form of the graphical analysis equation can be derived from the compartmental equations describing tracer accumulation in tissue 14, 21. For the two-tissue compartment model shown below the compartment equations are given in Eq. (1):
where C1 and C2 are concentrations (or radioactivities) for each compartment at time t. The units of radioactivity used in the examples presented in this paper are nCi/mL. The transfer constants
Distribution volume ratios using a reference region (without blood sampling)
The DVR can be calculated directly with the graphical method by using data from a reference region [REF(t)] with an average tissue-to-plasma efflux constant, k̄2 (to approximate the plasma integral) (17) [see Eq. (5)]: Solving for ∫0T Cpdt and substituting in Eq. (2), replacing k2REF with k̄2REF, gives Eq. (6): where int′ is int + δ, δ is the error term given by
Graphical analysis of irreversible ligands
Irreversibly binding ligands are essentially trapped for the time course of the scanning procedure. In terms of the two-compartment model pictured above, k4 = 0 so that tracer in C2 is trapped. Patlak et al. 20, 21 have shown that the rate constant (Ki) for the transfer of tracer from plasma to the irreversible compartment can be calculated from Eq. (8): which is linear for the times T > t′ when Ve, the distribution volume of the reversible part (the
Removing the bias in parameter estimates from linear methods
Although they present an advantage in ease of computation, the linearized equations can introduce a bias in the case of noisy data, since the error term at any given time point contains the error terms at the earlier time points. For example, the linear form of the one-tissue compartment model for scan times ti is [Eq. (14)]: with the equation errors, ξi, which are not statistically independent (7). This may result in biased parameter estimates. To
Conclusions
Graphical methods provide a quick, visual way to obtain information about the kinetics of tracer binding. In some cases these methods can be used without blood sampling if a suitable reference region is available. The reversible analysis requires scanning throughout the study to characterize the tissue uptake curve, which is necessary to evaluate the integral of the uptake. That graphical methods do not require a particular model structure is advantageous, since it is frequently the case that
Acknowledgements
Support was provided by Brookhaven National Laboratory under contract DE-AC02-98CH10886, by the U.S. Department of Energy and its Office of Biological and Environmental Research, and by the National Institutes of Health grant NS15380.
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