The Logan plot has been developed by Logan et al. [1] for ligands that bind reversibly to receptors and enzymes and is used for estimating the total distribution volume V_{T}. Its results can be interpreted with respect to the 1- and 2-tissue compartment models.

Operational Model Curve

The Logan plot belongs to a group of *Graphical Analysis* techniques, whereby the measured tissue TAC C_{T}(T) undergoes a mathematical transformation and is plotted against some sort of "normalized time". The Logan plot is given by the expression

with the input curve C_{p}(t). This means that the tissue activity integrated from the time of injection is divided by the instantaneous tissue activity, and plotted at a "normalized time" (integral of the input curve from the injection time divided by the instantaneous tissue activity). For systems with reversible compartments this plot will result in a straight line after an equilibration time t*.

In the derivation of the Logan plot the PET signal is described as a sum of tissue activity plus a fractional plasma signal

unlike the operational equation of the compartment model. Under these premises the slope represents the total distribution volume V_{T} plus the plasma space v_{P} in the VOI, which is usually neglected. Therefore

It has been found that the Logan plot is susceptible to noise in the data. Noise causes the true V_{T} to be underestimated, to a degree which not only depends on the noise level, but also on the local kinetics. The underestimation problem is particularly relevant in parametric mapping, where the pixelwise TACs suffer from a high noise level.

The reason for the underestimation effect is the fact that noise is not only present in the y-values (dependent variable) as the linear regression assumes, but also in the x-values (independent variable). To arrive at more accurate results it was therefore proposed to measure the residuals perpendicular to the regression line, rather than vertical to the x-axis [2].

Parameter Fitting

The **Logan Plot **model calculates and displays the measurements transformed as described by the formula above. It allows to fit a regression line within a range defined by the start time of the linear section **t***. The results are the distribution volume (slope) and the intercept. There is also an error criterion **Max Err** to fit **t***. For instance, if **Max Err** is set to 10% and the fit box of **t*** is checked, the model searches the earliest sample so that the deviation between the regression and all measurements is less than 10%. Samples earlier than the **t*** time are disregarded for regression and thus painted in gray. Note that **t*** must be specified in real acquisition time, although the x-axis units are in "normalized time". The corresponding normalized time which can be looked up in the plot is shown as a non-fitable result parameter **Start**.

The regression line is calculated using the traditional and the perpendicular distances, resulting in the total distribution volumes **Vt **and **Vt_perpend**, respectively. It has been found that **Vt** has a tendency to underestimate the distribution volume at increasing noise levels. **Vt_perpend** shows less bias due to noise, but suffers from a somewhat increased variability.

References

1. Logan J, Fowler JS, Volkow ND, Wolf AP, Dewey SL, Schlyer DJ, MacGregor RR, Hitzemann R, Bendriem B, Gatley SJ 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, 10(5):740-747. DOI

2. Varga J, Szabo Z: Modified regression model for the Logan plot. J Cereb Blood Flow Metab 2002, 22(2):240-244. DOI