# Patlak Reference Plot

## Introduction

Tracers undergoing irreversible trapping can be analyzed with the Patlak Reference Model. The Patlak plot [1] can be used as a reference model provided there is also some tissue where the tracer is not irreversibly bound. This is frequently applied in the analysis of [18F]-FDG studies, where a 2-tissue compartment model is used, with ${k}_{4}=0$

However, the Patlak Reference Model is also applicable to other compartment model situations, as long as there is one tissue compartment with irreversible trapping and a suitable reference tissue. Among other hypothesis, it is assumed that the distribution volume is the same for the tissue of interest (tissue including irreversible trapping) and the reference tissue: ${K}_{1}/{k}_{2}={K}_{1}^{\text{'}}/{k}_{2}^{\text{'}}$.

The operational equation for the model is:

$\frac{{C}_{\mathrm{tissue}}\left(t\right)}{{C}_{\mathrm{ref}}\left(t\right)}=K\frac{{\int }_{0}^{t}{C}_{\mathrm{ref}}\left(u\right)du}{{C}_{\mathrm{ref}}\left(t\right)}+V\text{}\left(1\right)\text{}$

Where ${C}_{\mathrm{tissue}}\left(t\right)$ is the TAC for the region including irreversible trapping and ${C}_{\mathrm{ref}}\left(t\right)$ is the TAC for the reference region.

In this approach, the measured PET activity ${C}_{\mathrm{tissue}}\left(t\right)$ is divided by the activity in the reference tissue. This value is plotted against the quotient at the other side of the equation: the integral of the TAC at the reference region divided by the activity in this same region. The latter acts as a sort of "normalized time". If the system has an irreversible compartment, this relation becomes linear after some time t* after injection, which allows estimating the slope and the intercept with a linear fit. The interpretation of these parameters depends on the particular configuration of the system.

When the compartment model in the figure is followed, the slope is:

$\mathrm{slope}=K=\frac{{k}_{2}{k}_{3}}{{k}_{2}+{k}_{3}}\text{}\left(2\right)$

The intercept is also a function of the transfer constants of the system, but it is not usually taken into account for the analyses.

## Input parameters

• TAC 1: TAC from a region of interest (region including irreversible trapping)
• TAC 2: TAC from a reference region
• Max. Err.: Maximum relative error allowed between the linear regression and the Patlak-transformed measurements $\left({C}_{\mathrm{tissue}}\left(t\right)/{C}_{\mathrm{ref}}\left(t\right)\mathrm{vs}.\text{}{\int }_{0}^{t}{C}_{\mathrm{ref}}\left(u\right)du/\text{}{C}_{\mathrm{ref}}\left(t\right)\right)$.
• t*: The linear regression estimation should be restricted to a range after an equilibration time. Parameter t* marks the beginning of the range used in the linear regression analysis. It can be fitted based on the Max. Err. criterion. If so, the model is fit with an initial t* starting at the beginning of the scan, and the estimation error is compared to the specified Max. Err. value. If the obtained error is larger, then the next possible t* is tried. The process continues until the obtained error is lower than the specified Max. Err. Alternatively, t* can be directly specified by the user. Parameter t* is an actual acquisition time value, even though a "normalized time" is used at the rigth-hand side of the equation.
• Threshold: Discrimination threshold for background masking.

## Output parameters and goodness of fit

• t* : It marks the beginning of the range of times used in the linear regression analysis. If the user does not directly specify t*, it is estimated by the program on the basis of a maximum allowed error. This happens at the preprocessing stage, when the model is fit to the mean TAC of a region with irreversible trapping. The value obtained in this way is then used as an input for the calculation of the parametric images.
• K : Slope of the linear regression.
• Intercept : Intercept of the linear regression.
• Start: "Normalized time" that corresponds to t*.
• Max. Error: Maximum relative error obtained between the linear regression and the Patlak-transformed measurements in the segment starting from 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] Patlak, C. S., Blasberg, R. G., & Fenstermacher, J. D. (1983). Graphical Evaluation of Blood-to-Brain Transfer Constants from Multiple-Time Uptake Data. Journal of Cerebral Blood Flow and Metabolism, 3(1), 1–7.

[2] Patlak, C. S., & Blasberg, R. G. (1985). Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations. Journal of Cerebral Blood Flow and Metabolism, 5(4), 584–590.