  Graphical analysis methods are widely used in positron emission tomography
quantification because of their simplicity and model independence. But they
may, particularly for reversible kinetics, lead to bias in the estimated
parameters. The source of the bias is commonly attributed to noise in the data.
Assuming a two-tissue compartmental model, we investigate the bias that
originates from model error. This bias is an intrinsic property of the
simplified linear models used for limited scan durations, and it is exaggerated
by random noise and numerical quadrature error. Conditions are derived under
which Logan's graphical method either over- or under-estimates the distribution
volume in the noise-free case. The bias caused by model error is quantified
analytically. The presented analysis shows that the bias of graphical methods
is inversely proportional to the dissociation rate. Furthermore, visual
examination of the linearity of the Logan plot is not sufficient for
guaranteeing that equilibrium has been reached. A new model which retains the
elegant properties of graphical analysis methods is presented, along with a
numerical algorithm for its solution. We perform simulations with the fibrillar
amyloid-beta radioligand [11C] benzothiazole-aniline using published data from
the University of Pittsburgh and Rotterdam groups. The results show that the
proposed method significantly reduces the bias due to model error. Moreover,
the results for data acquired over a 70 minutes scan duration are at least as
good as those obtained using existing methods for data acquired over a 90
minutes scan duration.
