# Patlak Reference Plot

## Contents

## 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.