Models and methods for derivation of in vivo neuroreceptor parameters with PET and SPECT reversible radiotracers
Introduction
The objective of a PET or SPECT neuroreceptor imaging study is to obtain quantitative information about the distribution of the target receptor throughout the brain. Several factors must be taken into account in order to form accurate inferences about receptor parameters. The activity recorded by the scanner will be due to some combination of specifically bound, non-specifically bound, and unbound radiotracer. The proportions of total measured activity represented by each of these pools are time varying and interdependent quantities. In addition to the actual distribution of receptors, peripheral clearance, regional cerebral blood flow and transport across the blood brain barrier will all influence the profile of activity over time. These factors can vary greatly from subject to subject and should be accounted for if results are to be adequately interpretable. Therefore, analysis of neuroreceptor imaging studies should not be limited to empirical outcome measures, such as the ratio of activity in a region of interest (ROI) to that in a region of reference (RREF) at one point in time. Model based methods relate the observed ROI time activity curves (the observed response function) to the arterial time activity curve (the observed input function) through a defined model (convolution with an unobservable but inferred impulse response function).
Model-based methods for in vivo quantification of neuroreceptors can be divided into kinetic, equilibrium and graphical methods. Kinetic methods yield quantitative information about the receptors from the estimation of the rate constants that characterize the tracer transfer between plasma, brain, and receptors [39]. Equilibrium methods derive this information from analysis of the activity distribution at equilibrium (i.e., when the receptor-ligand association and dissociation rates are equal) [13], [28]. In the graphical approach, data are transformed to new variables that are linearly related, and the parameters of interest are derived by linear regression [36], [44]. In this paper, we will look at several formulations in each of these categories. While our choices are not intended to comprise an exhaustive list, we do believe they are representative of the range of analysis techniques in current use.
Section snippets
Compartments and fractional rates constants
The notion of compartment is essential to most model based methods. A compartment is a physiological or biochemical “space” in which the tracer concentration (C(t)) is assumed to be homogeneous at all times. One of the most general compartment configurations used to describe radiotracer brain uptake and binding is depicted in Fig. 1 [15]. In a region with specific binding sites, 4 compartments are described: the plasma compartment (C1), the intracerebral compartment in which the tracer is free
Determination of receptor parameters
The standard compartment model has been widely accepted as a paradigm for the underlying kinetics in PET radiotracer experiments. The parameter estimation process consists of finding a set of rate constants that produce a model curve Ca(t) ⊗ h(t) which is as close as possible to the measured data in some prescribed sense. The three general methods based on compartment models—kinetic, equilibrium and graphical—each have variants in which the input function is either measured directly as the
Limitations of SRTM
With several tracers, SRTM has produced results that are discrepant with conventional kinetic modeling [6], [18], [41]. We and others have performed simulations [4], [47], [48] showing that the simplification step, along with the intrinsic way in which the assumed two compartment configuration in the reference region enters the equations for SRTM, can cause the method to produce estimates that are biased. The magnitude of the bias is dependent on the magnitude of the parameters themselves. For
The graphical method of Logan et al. with arterial plasma input function
Logan et al. [36] proposed an approach to modeling of reversible tracers which is similar in spirit to Patlak’s method for irreversible (i.e., k4 = 0) tracers [44]. In this method, a transformation of variables is performed in which the transformed dependent and predictor variables have an asymptotically linear relationship. The slope of the asymptote is VT. The data rapidly approach the asymptote (Fig. 3) so VT can be estimated from the slope of the nearly linear part of the graph. The
Equilibrium methods
The information to be inferred from PET radioligand experiments pertains to the relationship between brain and ligand under equilibrium conditions. Yet none of the methods presented thus far require the system to be in equilibrium in order to estimate the relevant parameters. Rather, they rely upon mathematical models to infer equilibrium parameters from data acquired while concentrations are continuously changing. Several approaches exist in which measurements are made at equilibrium. In the
Conclusion
In this article we have presented a number of different approaches to estimation of the outcome measures V3, V′3 and V″3. All the methods start with similar or even identical modeling assumptions. While the parameter estimates given by all the methods generally correlate well (for example, all will usually provide the same rank order of regional binding) differences (occasionally large differences) have been observed. Kinetic modeling is the most direct implementation of the compartment models,
Acknowledgements
Supported by US Public Health Service (K02-MH01603-01, RO1-MH54192, ROI MH59144-01, and MH59342-01). The authors thank Dr. Anissa Abi-Dargham for the data used to generate Fig. 3.
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