# Dual time point estimation of *K*_{i}

_{i}

Net influx rate (*K _{i}*) of a PET tracer is usually
estimated from a dynamic PET data using either Patlak plot or
compartmental model, and it requires the measurement of
arterial plasma curve starting from the tracer administration until the end of the PET scan.
Blood sampling is not necessary, if the
input function can be measured from the dynamic PET image.
An estimate of

*K*, FUR, does not require dynamic imaging, but can be calculated from a static late-scan image; however, FUR calculation still needs the input function (integral) from the whole time, starting from the tracer administration until the end of the PET scan, and late-scan image can not provide that. If the concentration in the blood can be measured during the late-scan, then tissue-to-blood ratio can be calculated, and it is superior to SUV as a surrogate parameter of

_{i}*K*(van den Hoff et al., 2013).

_{i}If two static late-scans can be performed, and
input function can be measured from the PET images, then
*K _{i}* (and subsequently metabolic rate) can be
calculated from that dual time point (DTP) data alone
(van den Hoff et al., 2013), assuming that

- there is an irreversible compartment in tissue in which the tracer is trapped (Patlak plot and FUR methods have this same requirement), and
- input function follows mono-exponential decay between the two PET scans, and
- a population-based estimate of the y axis intercept of Patlak plot is available (this would also improve the FUR method).

If the tracer concentration in plasma and tissue at time *t* are represented by
*C _{P}(t)* and

*C*, respectively, and the PET scan times are

_{T}(t)*t*and

_{1}*t*, then the estimate of

_{2}*K*can be calculated from equation:

_{i}, where *Ī _{r}* is the population-based estimate of the y axis intercept of
the Patlak plot.

Alternatively, if the time course of the input function is assumed to follow an inverse power law
(hyperbola) for *t* ≳ 1-2 min *p.i.*,

, then even simpler equation for an estimate of *K _{i}* can be derived
(Hofheinz et al., 2016):

, but usage of this method requires that the inter- and intra-individual variability of
*b* can be ignored and a population average of *b* has been calculated.

If input function is available from the injection time, dual-time point *K _{i}*
can be calculated from equation

, from which the intercept of the Patlak plot has been cancelled out (Wu et al., 2021). If input function is not measured, but blood radioactivity concentration can be assessed from the two PET scans, then those values could be used to scale population-based input function to individual level (Wu et al., 2021).

## DTP with reference tissue

If arterial plasma data at the times of the two PET scans is not available, not even from the
images, but a reference
region can be found in the images, then it can be used to calculate a surrogate parameter
for *K _{i}^{ref}*, that in case of
dynamic imaging could be calculated using Patlak plot with reference tissue input.
Alves et al (2017) have applied this method
to [

^{18}F]FDOPA studies, using occipital cortex as reference region:

*C _{R}* represents the radioactivity concentration in the reference tissue as a
function of time.

*RPT*(reference Patlak time) is the area-under-curve (AUC) of the reference tissue from tracer administration time (

*t=0*) to the scan time (

*t*or

_{1}*t*) divided by the concentration in the reference region at the corresponding scan time. For the method to be useful, the

_{2}*RPT(t*must be assumed to have sufficiently low inter- and intra-individual variability, so that a population average of

_{1})*RPT(t*can be calculated and used in the previous equation, and in calculation of

_{1})*RPT(t*from equation

_{2})## See also:

- Patlak plot from regional TACs with plasma input
- Patlak plot from regional TACs with reference tissue input
- Calculation of Patlak plot for dynamic images
- Fractional Uptake Rate (
*FUR*) - Retention index (
*RI*) - Static late scan
- Late-scan tissue-to-plasma ratio

## Literature

Alves LI, Meles SK, Willemsen AT, Dierckx RA, Marques da Silva AM, Leenders KL, Koole M.
Dual time point method for the quantification of irreversible tracer kinetics: A reference tissue
approach applied to [^{18}F]-FDOPA brain PET.
*J Cereb Blood Flow Metab.* 2017; 37(9): 3124-3134.
doi: 10.1177/0271678X16684137.

van den Hoff J, Hofheinz F, Oehme L, Schramm G, Langner J, Beuthien-Baumann B, Steinbach J,
Kotzerke J. Dual time point based quantification of metabolic uptake rates in ^{18}F-FDG PET.
*EJNMMI Res.* 2013; 3: 16.
doi: 10.1186/2191-219X-3-16.

Hofheinz F, van den Hoff J, Steffen IG, Lougovski A, Ego K, Amthauer H, Apostolova I.
Comparative evaluation of SUV, tumor-to-blood standard uptake ratio (SUR), and dual time point
measurements for assessment of the metabolic uptake rate in FDG PET.
*EJNMMI Res.* 2016; 6(1): 53.
doi: 10.1186/s13550-016-0208-5.

Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers.
*Nucl Med Biol.* 2000; 27: 661-670.
doi: 10.1016/S0969-8051(00)00137-2.

Patlak CS, Blasberg RG. Graphical evaluation of blood-to-brain transfer constants from
multiple-time uptake data. Generalizations. *J Cereb Blood Flow Metab.* 1985; 5: 584-590.
doi: 10.1038/jcbfm.1985.87.

Tags: Patlak plot, Ki, Late-scan

Updated at: 2022-01-14

Created at: 2018-02-14

Written by: Vesa Oikonen