Graphical analysis of PET data applied to reversible and irreversible tracers

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Abstract

The differential equations of compartmental analysis form the basis of the models describing the uptake of tracers used in imaging studies. Graphical analyses convert the model equations into linear plots, the slopes of which represent measures of tracer binding. The graphical methods are not dependent upon a particular model structure but the slopes can be related to combinations of the model parameters if a model structure is assumed. The input required is uptake data from a region of interest vs time and an input function that can either be plasma measurements or uptake data from a suitable reference region. Graphical methods can be applied to both reversible and irreversibly binding tracers. They provide considerable ease of computation compared to the optimization of individual model parameters in the solution of the differential equations generally used to describe the binding of tracers. Conditions under which the graphical techniques are applicable and some problems encountered in separating tracer delivery and binding are considered. Also the effect of noise can introduce a bias in the distribution volume which is the slope of the graphical analysis of reversible tracers. Smoothing techniques may minimize this problem and retain the model independence. In any case graphical techniques can provide insight into the binding kinetics of tracers in a visual way.

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): dC1dt=K1Cp(t)−(k2+k3)C1+k4C2

dC2dt=k3C1−k4C2

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, 2 (to approximate the plasma integral) (17) [see Eq. (5)]: 0T REF(t) dtREF(T)0T Cp(t) dtREF(T)1k2REF where λ=K1REF/k2REF Solving for ∫0T Cpdt and substituting in Eq. (2), replacing k2REF with 2REF, gives Eq. (6): 0T ROI(t) dtROI(t)=DVR 0T REF(t) dt+REF(T)/k̄2REFROI(T) +int′ 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): ROI(T)Cp(T)=Ki 0T Cp(t)dtCp(T)+(Ve+Vp) 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)]: C(ti)=K1 0ti Cpdt−k2 0ti C1(t)dt+ξi 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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