BPnd (Logan Ref): Logan Reference Plot with fixed k2'

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BPnd (Logan Ref): Logan Reference Plot with fixed k2'

Logan et al. [1] developed a reference tissue method for reversible receptor ligands which does not depend on a specific model structure of the reference tissue. Assuming the presence of reference region TAC CT'(t) with an average tissue-to-plasma clearance  k2', the target tissue TAC CT(t) is transformed and plotted as a function of the transformed reference TAC, as illustrated below.


The operational equation resembles a linear equation with the distribution volume ratio (DVR = BPND+1) as the slope plus an error term which decreases over time. Therefore the late part starting from a time t* of the plotted samples can be fitted by a regression line and the slope used for calculating BPND. The time t* can be determined as the time after which no further significant increases in slope are observed.

The graphical plot of the Logan Reference Tissue method is described by the following equation with the form resembling a linear equation.

Equation Logan Noninvasive

k2' in the original publication was the population average k2 determined for the reference tissue using blood sampling, but using the subject's own k2' may be preferable.

Acquisition and Data Requirements

Image Data

A dynamic data set acquired long enough that the equilibrium relation is approximately fulfilled.

Target tissue

Optional: TAC from a receptor-rich region (such as basal ganglia for D2 receptors).

Reference tissue

Mandatory: TAC from a receptor-devoid region (such as cerebellum or frontal cortex for D2 receptors).

Note: specification of an appropriate reference TAC is crucial for the result!


Optional: VOI definition excluding the reference tissue which can be used for getting an estimate of k2'.

Model Preprocessing

PXMOD Logan Non-Invasive Model Pre-processing


The linear regression estimation should be restricted to a range after an equilibration time. t* marks the beginning of the range used in the linear regression analysis. It can be fitted based on the Max. Err. criterion.
Note: Time is in acquisition time, not in the normalized time of the graphical plot.

Max. Err.

Maximum relative error allowed between the linear regression and the Logan-transformed measurements in the segment starting from t*.

Percent masked pixels

Exclude the specified percentage of pixels based on histogram analysis of integrated signal energy. Not applied in the presence of a defined mask.

k2' for mapping

k2 of the reference tissue. It can be specified in four different ways, as explained in section Specification of k2'.

The result of a model fit during Model Preprocessing is shown in the Model Results panel for inspection. If no Target tissue is specified, the panel remains empty.

PXMOD Logan Non-Invasive Model Pre-processing

Model Configuration

PXMOD Logan Non-Invasive Model Parameters


Binding potential, calculated by as: BPnd = k3/k4 = DVR-1.0 where DVR is the slope of the fitted regression line.


Intercept of the linear regression (Logan Eq.6.)


1.Logan J, Fowler JS, Volkow ND, Wang GJ, Ding YS, Alexoff DL: Distribution volume ratios without blood sampling from graphical analysis of PET data. J Cereb Blood Flow Metab 1996, 16(5):834-840. DOI