# Logan Reference Plot with fixed ${k}_{2}^{\text{'}}$

## Introduction

The Logan Reference Tissue method is used to estimate the distribution volume ratio ($\mathrm{DVR}$) for PET studies with reversible radioligands. $\mathrm{BP}$ is then derived from $\mathrm{DVR}$ as $\mathrm{BP}=\mathrm{DVR}-1$.

The method does not depend on a specific model structure of the reference tissue. Assuming the presence of reference region TAC ${C}_{{T}^{\text{'}}}\left(t\right)$ with an average tissue-to-plasma clearance ${k}_{2}^{\text{'}}$, the target tissue TAC ${C}_{T}\left(t\right)$ is transformed and plotted as a function of the transformed reference TAC, which acts as a sort of "normalized time".

The operational equation for the method is:

$\frac{{\int }_{0}^{t}{C}_{T}\left(\tau \right)d\tau }{{C}_{T}\left(t\right)}=\mathrm{DVR}\frac{{\int }_{0}^{t}{C}_{{T}^{\text{'}}}\left(\tau \right)d\tau +{C}_{{T}^{\text{'}}}\left(t\right)/{k}_{2}^{\text{'}}}{{C}_{T}\left(t\right)}+b$

This resembles a linear equation, where $\mathrm{DVR}$ is the slope and $b$ is an error term which decreases over time. This dependence also becomes negligible after some time t*. Then, if only the values obtained after t* are taken into account, a linear regression allows estimating $\mathrm{DVR}$ and $b$.

As mentioned, ${k}_{2}^{\text{'}}$ represents the average tissue to plasma efflux. It has to be specified as an input for the model. A representative value for the population based on previous studies can be entered, or, alternatively, ${k}_{2}^{\text{'}}$ can be estimated for each subject by applying, for instance, the SRTM method, before starting the Logan analysis.

## Input parameters

• TAC 1: TAC from a region with specific uptake (typically a receptor rich area).
• TAC 2: TAC from a reference region with no specific uptake (typically, a region devoid of target receptors).
• t*: 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. The graphical representation of the data provided by the program can allow the user to select appropriate values for t* based on a visual analysis. This parameter can also be fitted automatically using the Max. Err. criterion. In this case, the lowest t* yielding an error lower than Max. Err. (see below) will be chosen. Note that the t* is specified in acquisition time units. The program later translates this time into the units of the x axis (the "normalized time") and shows it as the Start parameter.
• Max. Err.: Maximum relative error allowed between the linear regression and the Logan Plot-transformed measurements in the segment starting from t*.
• Threshold: Discrimination threshold for background masking.
• ${k}_{2}^{\text{'}}$: an average value of the efflux rate constant from regions without receptors, which has been previously determined.

## Output parameters and goodness of fit

• BP: Binding potential obtained as $\mathrm{BP}=\mathrm{DVR}-1$ from the slope of the linear regression.
• Intercept :Intercept of the linear regression.
• Start: The value of the x axis corresponding to the time point t*.

- Goodness of fit:

To show the goodness of fit at the preprocessing step, two parameters are given, together with the estimated parameters:

• Normalized Mean Squared Error (NMSE):

$\mathrm{NSME}=\frac{||{C}_{t}-{C}_{{t}_{\mathrm{estimate}}}{||}^{2}}{||{C}_{t}-\mathrm{mean}\left({C}_{t}\right){||}^{2}}$

It measures the quality of the fit for the TACs. Values vary between $-\infty$ (bad fit) to 1 (perfect fit).

• Correlation coefficient (Corr. Coef.): The correlation coefficient between the values of the TAC of interest and the values of the TAC estimated by the model. Values closer to 1 are better.

## Main references

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