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Standard Logan Plot, Abstract [21]
"A graphical method of analysis applicable to ligands that bind reversibly to receptors or enzymes requiring the simultaneous measurement of plasma and tissue radioactivities for multiple times after the injection of a radiolabeled tracer is presented. It is shown that there is a time t after which a plot of integral of ROI(t')dt'/ROI(t) versus integral of Cp(t')dt'/ROI(t) (where ROI and Cp are functions of time describing the variation of tissue radioactivity and plasma radioactivity, respectively) is linear with a slope that corresponds to the steady-state space of the ligand plus the plasma volume,.Vp. For a two- compartment model, the slope is given by lambda + Vp, where lambda is the partition coefficient and the intercept is -1/[kappa 2(1 + Vp/lambda)]. For a three-compartment model, the slope is lambda(1 + Bmax/Kd) + Vp and the intercept is -[1 + Bmax/Kd)/k2 + [koff(1 + Kd/Bmax)]-1) [1 + Vp/lambda(1 + Bmax/Kd)]-1 (where Bmax represents the concentration of ligand binding sites and Kd the equilibrium dissociation constant of the ligand-binding site complex, koff (k4) the ligand-binding site dissociation constant, and k2 is the transfer constant from tissue to plasma). This graphical method provides the ratio Bmax/Kd from the slope for comparison with in vitro measures of the same parameter. It also provides an easy, rapid method for comparison of the reproducibility of repeated measures in a single subject, for longitudinal or drug intervention protocols, or for comparing experimental results between subjects. Although the linearity of this plot holds when ROI/Cp is constant, it can be shown that, for many systems, linearity is effectively reached some time before this. This analysis has been applied to data from [N-methyl-11C]-(-)-cocaine ([11C]cocaine) studies in normal human volunteers and the results are compared to the standard nonlinear least- squares analysis. The calculated value of Bmax/Kd for the high-affinity binding site for cocaine is 0.62 +/- 0.20, in agreement with literature values."
Logan Plot with Perpendicular Distances in Regression, Abstract [32]
"Logan's graphical model is a robust estimation of the total distribution volume (DVt) of reversibly bound radiopharmaceuticals, but the resulting DVt values decrease with increasing noise. The authors hypothesized that the noise dependence can be reduced by a linear regression model that minimizes the sum of squared perpendicular rather than vertical (y) distances between the data points and fitted straight line. To test the new method, 15 levels of simulated noise (repeated 2,000 times) were added to synthetic tissue activity curves, calculated from two different sets of kinetic parameters. Contrary to the traditional method, there was no ( P > 0.05) or dramatically decreased noise dependence with the perpendicular model. Real dynamic 11C (+) McN5652 serotonin transporter binding data were processed either by applying Logan analysis to average counts of large areas or by averaging the Logan slopes of individual-voxel data. There were no significant differences between the parameters when the perpendicular regression method was used with both approaches. The presented experiments show that the DVt calculated from the Logan plot is much less noise dependent if the linear regression model accounts for errors in both the x and y variables, allowing fast creation of unbiased parametric images from dynamic positron-emission tomography studies."
Ichise's bilinear Method MA1 [33]
"In an attempt to improve neuroreceptor distribution volume (V) estimates, the authors evaluated three alternative linear methods to Logan graphical analysis (GA): GA using total least squares (TLS), and two multilinear analyses, MA1 and MA2, based on mathematical rearrangement of GA equation and two-tissue compartments, respectively, using simulated and actual PET data of two receptor tracers, [(18)F]FCWAY and [(11)C]MDL 100,907. For simulations, all three methods decreased the noise-induced GA bias (up to 30%) at the expense of increased variability. The bias reduction was most pronounced for MA1, moderate to large for MA2, and modest to moderate for TLS. In addition, GA, TLS, and MA1, methods that used only a portion of the data (T > t*, chosen by an automatic process), showed a small underestimation for [(11)C]MDL 100,907 with its slow kinetics, due to selection of t* before the true point of linearity. These noniterative methods are computationally simple, allowing efficient pixelwise parameter estimation. For tracers with kinetics that permit t* to be accurately identified within the study duration, MA1 appears to be the best. For tracers with slow kinetics and low to moderate noise, however, MA2 may provide the lowest bias while maintaining computational ease for pixelwise parameter estimation."