A review of graphical methods for tracer studies and strategies to reduce bias

https://doi.org/10.1016/S0969-8051(03)00114-8Get rights and content

Abstract

Graphical techniques provide simple methods for the analysis of data from tracer studies. 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. The theoretical work of Patlak which was applied to irreversible tracers formed the basis for extensions of graphical techniques to reversibly binding tracers. The advantage of graphical methods is that they are not dependent upon a particular model structure but provide a measure of tracer binding that can be interpreted in terms of a model structure if desired. They provide a visual way to distinguish the type of binding whether reversible or irreversible in the initial studies of new ligands. Conditions under which the graphical techniques can be applied are considered as well as problems encountered with slow binding components. One problem in the use of these methods particularly the method for reversible tracers is the bias generated due to the presence of statistical noise. Some recently proposed techniques for reducing the noise are considered.

Introduction

PET and SPET cameras record the time variation of labeled tracer within the body after its introduction generally as a bolus injection. In order to convert this time sequence of radioactivities into one or more numbers that are related to actual physiological processes, it is necessary to apply compartmental models. These models are certainly oversimplifications of the true physiology but nevertheless they allow the estimation of model parameters for the comparison of subjects under different experimental conditions and/or belonging to different groups. In particular, models allow the separation of processes of primary interest such as free receptor concentration from other processes related to tracer uptake. Frequently parameter estimation is done by an iterative nonlinear least squares method requiring a measured plasma input function in which radioactivity of the tracer is measured separately from the metabolites. The differential equations of the compartment models can be linearized and the model parameters can be estimated using standard techniques for linear equations without the requirement of an iterative search (for example the work of Evans and Blomqvist [2], [3]). Graphical analysis is a further simplification of general linear methods - it converts the model equations into one equation evaluated at the time points corresponding to the scanning times and provides fewer parameters, namely a slope and intercept. The graphical methods are independent of any particular model structure, although the slope can be interpreted in terms of a combination of model parameters for some model structure. Graphical methods also require an input function although in some instances a reference region can be used in place of plasma input if it is devoid of the specific binding sites. Graphical analysis (GA) methods have been developed for reversibly [4], [5], [6], [7] and irreversibly binding tracers [1], [8], [9]. In the case of irreversible tracers some fraction of the radioactivity is trapped at the binding site for the duration of the experiment. Reversible tracers on the other hand demonstrate uptake and loss from all compartments over the time of the study. The method for irreversible tracers was developed first - the theoretical foundation was provided by Patlak [1], [9]. The extension to reversible systems developed by Logan [4] was based on the original work of Patlak. Further refinements have been made by Ichise [7]. The main problem with the use of graphical methods is the bias in the estimated parameters due to noise [10]. Recently some methods for dealing with noise have been proposed [11], [12]. We present here a review of the graphical methods, and recent developments in strategies to improve parameter estimation by reducing the bias.

Section snippets

Graphical analysis of reversible tracers

In general the compartmental equations can be written using Patlak's notation, [1], [9] dC̃dt=K̃C̃+Q̃Cp(t) where C̃ is the vector of compartmental concentrations at time t, K̃ is a matrix of the transfer constants between compartments, Q̃ is a vector of plasma to tissue transfer constants which generally consists of one nonzero component designated as K1. Cp(t) is plasma concentration of unmetabolized tracer (the input function). Using ROI(t)=ŨnτC̃+Vp=i Ci(t)Vp where ROI(t) (region of

Distribution volume ratio

Generally the DV ratio (DVR) or binding potential, which can be derived from the DVR are used for comparing data sets. The DVR is the ratio of the DV in a receptor region to that of a reference region. This generally provides better reproducibility on test/retest than comparing either the DV or the receptor parameter k3. If the DV of the reference region has one tissue compartment, then the binding potential (BP) can be derived from the DVR DVR=λROIλREF(1+BP) where λ=K1/k2 for ROI or REF, if

Graphical analysis using a reference region

We can extend graphical analysis to obtain DV ratios directly without blood sampling by using a reference region in place of the plasma integral. This can be done by rearranging the graphical analysis equation for the reference region to solve for the plasma integral in terms of the reference region radioactivity. Rearranging Eq. (6a) where REF is used in place of ROI gives 0tCp(t′)dt=1DVREF 0tREF(t′)dt−intREFREF(t) . Substituting for the integral of the plasma in the equation for the

Graphical analysis of irreversible tracers

Some tracers bind irreversibly, for example the monoamine oxidase (MAO) tracers [11C] L-deprenyl, and the deuterium substituted [11C] L-deprenyl-D2 [16], [17], [18] are suicide inhibitors for MAO B and similarly [11C] clorgyline and [11C]clorgyline-D2 irreversibly inhibit MAO A [19]. Other tracers may appear irreversible over the course of the PET experiment because the ligand receptor dissociation is very slow. In addition to Eq. (1) which describes the reversible parts of the system, the

Some problems with irreversible tracers

A two tissue compartment model with one compartment representing the trapped component as illustrated below Cp(t) K1k2 C1(t) k3 C2(t) has Ki=Kik3k2+k3=K1λk3K1+λk3 . The ratio K1/k2 is not a function of blood flow [20], however, Ki depends upon blood flow since K1 is a function of blood flow and capillary permeability [21]. An increase in Ki could be due to either an increase in blood flow or an increase in tracer binding. Since it is usually the change in tracer binding that is of primary

Using a plasma input function

The differential equations used to describe movement of tracer between plasma and tissue compartments can be linearized so that the model parameters are more easily determined by solving a set of linear equations [2], [3]. For example the differential equation for one tissue compartment model dC1(t)dt=K1Cp(t)−k2C1 becomes C(ti)=K10tiCp(t)dt+k20tiC1(t)dt+ξi where ξi are the error terms. Since the error terms are not statistically independent, the parameter estimates may be biased [22], [23].

Using a reference region input

As with the plasma input function form of the GA equation, the reference region method is also susceptible to noise. Using simulated ROI data without noise both reference region methods give very similar values for the DVR, 4.44 and 4.41 for the bilinear and GA respectively (the “true” value is 4.43mL/mL). For comparison the Simplified reference tissue method (SRTM) [27] gives 4.68 for the DVR. The SRTM assumes that both receptor and reference region can be adequately described by a 1 tissue

Some problems with reversible tracers

11C cocaine binds reversibly to the dopamine transporter and has been used in a number of studies [28], [29], [30]. In recent baboon studies we have encountered poor reproducibility on test/retest which seems to be related to some extent to blood flow. Studies done under different conditions of ventilation resulted in very large differences in blood flow and this appears to affect the value of the DV. In order for blood flow to have an effect on the DV estimate, implies that the DV is being

Conclusions

The graphical analysis methods originated with the treatment of irreversibly binding tracers. The elegant theoretical work of Patlak [1], [9] established the conditions under which the GA method is applicable and was the starting point for the derivation of the method for reversible tracers. The methods are simple to apply and can provide information by a simple plot (Eq. (6a) and Eq. (9) about the type of binding, that is whether it is reversible or irreversible without the necessity of

Acknowledgements

This work was supported by U.S. D.O.E (OBER; DE-AC02-98CH10886) and N.I.H. (NS-15380).

References (31)

  • Y. Kimura et al.

    Improved signal-to-noise ratio in parametric images by cluster analysis

    Neuroimage

    (1999)
  • A.A. Lammertsma et al.

    Simplified reference tissue model for PET receptor studies

    Neuroimage

    (1996)
  • C.S. Patlak et al.

    Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data

    J Cereb Blood Flow Metab

    (1983)
  • A.C. Evans

    A double integral form of the three compartment, four rate-constant model for faster generation of parametric maps

    J Cereb Blood Flow Metab

    (1987)
  • G. Blomqvist

    On the construction of functional maps in positron emission tomography

    J Cereb Blood Flow Metab

    (1984)
  • J. Logan et al.

    Graphical analysis of reversible radioligand binding from time-activity measurements applied to [N-11C-methyl]-(-)-cocaine PET studies in human subjects

    J Cereb Blood Flow Metab

    (1990)
  • J. Logan et al.

    Distribution volume ratios without blood sampling from graphical analysis of PET data

    J Cereb Blood Flow Metab

    (1996)
  • M. Ichise et al.

    From graphical analysis to multilinear regression analysis of reversible radioligand binding

    J Cereb Blood Flow Metab

    (1996)
  • M. Ichise et al.

    Graphical analysis and simplified quantification of striatal and extrastriatal dopamine D2 receptor binding with [123I]epidepride SPECT

    J Nucl Med

    (1999)
  • A. Gjedde

    High-, and low-affinity transport of D-glucose from blood to brain

    J Neurochem

    (1981)
  • C.S. Patlak et al.

    Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations

    J Cereb Blood Flow Metab

    (1985)
  • M. Slifstein et al.

    Effects of statistical noise on graphic analysis of PET neuroreceptor studies. Journal of Nuclear Medicine : Official Publication

    Soc Nuc Med

    (2000)
  • M. Ichise et al.

    Strategies to improve neuroreceptor parameter estimation by linear regression analysis

    J Cereb Blood Flow Metab

    (2002)
  • J. Logan et al.

    A strategy for removing the bias in the graphical analysis method

    J Cereb Blood Flow Metab

    (2001)
  • M. Mintun et al.

    A quantitative model for the in vivo assessment of drug binding sites with positron emission tomography

    Ann Neurol

    (1984)
  • Cited by (82)

    • Multimodal imaging of capsid and cargo reveals differential brain targeting and liver detargeting of systemically-administered AAVs

      2022, Biomaterials
      Citation Excerpt :

      The biodistribution after perfusion was assessed at 22 h after injection of radiolabeled capsids, and the enhanced brain accumulation of the novel capsids was confirmed (Fig. S3A). A Logan plot is a graphical analysis method for the estimation of the reversible uptake of tracer [34]. Logan plots confirmed that the 30-min accumulation in the brain was greater for 64Cu-PHP.eB than 64Cu-CAP-B10 and much greater than 64Cu-AAV9, with a distribution volume of 0.207, 0.143, and 0.0227, respectively (Fig. 1J).

    • Parametric imaging used in nuclear medicine

      2022, Nuclear Medicine and Molecular Imaging: Volume 1-4
    View all citing articles on Scopus
    View full text