# Adding weights to PET TAC data

The time frames of regional tissue time-activity curves (TTACs) have different noise statistics, and the sample frequency is usually higher in the early frames (frames are shorter) in order to record precisely the kinetics of the rapidly changing activity after bolus administration of the radioligand. In nonlinear least-squares (NLLS) fitting the data samples may need to be weighted in order to cancel these effects and to get a decent fit to the whole time range of the TTAC (Yaqub et al., 2006).

Weighting may also be necessary to properly fit plasma TACs. In contrast to the PET scanner and online blood sampling systems, the radioactivity of manually drawn of plasma samples is usually measured using pre-defined count limit, that is, about the same number of counts is collected whether the radioactivity is low or high, and the collection time is varied. Therefore the variance in blood or plasma input curves (BTAC or PTAC) cannot be estimated based on the concentration. Generally, only the effect of decay correction needs to considered. However, the samples are taken less frequently towards the end of the PET scan, which should be compensated for in the weighting.

## Weighting model for regional TTACs

Weights can be based on a variance, σ^{2}, that can be estimated for each
individual time frame *i* from equation
(Jovkar et al., 1989;
Chen et al., 1991):

, where *C _{T}(t_{i})* is the radioactivity concentration at the mid-time
of frame, and

*Δt*is the frame length. For simulation of noise an additional proportionality coefficient will be needed. Weight is then calculated as:

_{i}If the data to be weighted is corrected for physical decay
(as is usual), then *C _{T}(t_{i})* in this equation should as well be
decay corrected.

However, variance estimation should not be based on individual regional or image pixel values,
but on the number of events collected during the time frame
(Mazoyer et al., 1986).
Based on sinogram total events, *c _{i}*,

, or for weighting decay corrected data (Wu et al., 2016; variance model 1 by Yaqub et al., 2006):

, where *f _{dc}* is the decay correction factor.

Noise distribution can be assumed uniform in the image even when the radioactivity concentration is heterogeneous (Asselin et al., 2004).

Since there is no generally agreed method to quantify statistical
noise in PET images, many
alternative weighting schemes have been proposed, such as using time frame lengths as weights.
Thiele & Buchert (2008) proposed
reducing the weight of the late frames because of the decay, using
the half-life of the radiolabel, *T _{1/2}* in the equation:

The noise in input function should be considered in the fitting of compartmental models to dynamic data (Huesman, 1997).

## Procedure

Most in-house model analysis programs can read the weights that are added to DFT files, or can optionally calculate the weights during program execution.

Weights can be added to tissue data file using dftweigh. Weights can be set based on either SIF data, or if SIF does not exist, based on the average tissue curves. The added weights are not absolute, but only relational to each time frame in the TAC.

Examples:

If weights are based on the SIF, the command is:

dftweigh ua5268.dft ua5268dy1.img.sif

If SIF file is not used, the command is simply:

dftweigh ua5268.dft

Note that weights need to be added to the DFT datafile only once, and they may affect the results
of other calculation programs. The weights can be removed using the same program,
`dftweigh`

, with option `-rm`

.

Different weighing methods can be optionally selected with
dftweigh; the default method
(Mazoyer et al., 1986) is
widely used and also proposed to be used with
SRTM, but a simpler weighting method that only
considers the time frame durations and physical decay (option `-wfd`

) was found to
provide optimal results by Thiele et al
(2008).

## See also:

## References:

Hughes IG, Hase TPA: *Measurements and their Uncertainties - A Practical Guide to Modern Error
Analysis*. Oxford University Press, 2010. ISBN 978-0-19-956632-7.

Mazoyer BM, Huesman RH, Budinger TF, Knittel BL. Dynamic PET data analysis.
*J Comput Assist Tomogr.* 1986; 10: 645-653. doi:
10.1097/00004728-198607000-00020.

Muzic RF Jr, Christian BT. Evaluation of objective functions for estimation of kinetic
parameters. *Med Phys.* 2006; 32(2): 342-353.
doi: 10.1118/1.2135907.

Normandin MD, Koeppe RA, Morris ED. Selection of weighting factors for quantification of PET
radioligand binding using simplified reference tissue models with noisy input functions.
*Phys Med Biol.* 2012; 57: 609-629.
doi: 10.1088/0031-9155/57/3/609.

Thiele F, Buchert R. Evaluation of non-uniform weighting in non-linear regression for
pharmacokinetic neuroreceptor modelling. *Nucl Med Commun.* 2008; 29: 179-188.
doi: 10.1097/MNM.0b013e3282f28138.

Yaqub M, Boellaard R, Kropholler MA, Lammertsma AA. Optimization algorithms and weighting
factors for analysis of dynamic PET studies. *Phys Med Biol.* 2006; 51: 4217-4232.
doi: 10.1088/0031-9155/51/17/007.

Tags: Weighting, Noise, Variance, Fitting, NLLS

Updated at: 2019-01-12

Created at: 2014-05-17

Written by: Vesa Oikonen, Lars Jødal