# Comparing PET TACs

In the analysis of dynamic PET data we often need to test whether two time-activity curves (TACs) are similar. When compartmental model or mathematical functions are fitted to measured, noisy TACs, we need to visually verify the goodness of the fit, that is, the fitted and measured TACs should be similar, except for the noise. The comparison is preferably done by plotting residuals (fitted value - measured value) as a function of time: residuals should oscillate randomly below and above zero. In image clustering and segmentation we must determine whether TACs of two image voxels have similar kinetics, and can be lumped together. For defining ROIs on the PET image, it may be useful to enhance differences in TAC kinetics for instance to find blood vessels.

Visual analysis of plotted TACs or TAC differences is not feasible in processing image voxel data, and too time consuming to be routinely used in comparison of ROI or input TACs. Direct statistical tests (paired t-test, ANOVA) between measured data points from the two curves is not recommended (Motulsky & Christopoulos, 2004). Runs test, or its simplest and fastest form, maximum run length, can be used to test the similarity of two curves.

In clustering and segmentation of dynamic PET images, traditional method has been to normalize
individual images to the administered dose and
possibly individual mass, fat-free mass, or body-surface are, and then calculate mean TACs of
necessary tissue types. Likelihood for a voxel to belong to each tissue type is then calculated as
Mahalonobis distance *D _{M}*

, where *p(t)* is the voxel value at time frame *t*, and *μ(t)* and
*σ(t)* are the mean and standard deviation of the tissue type, respectively.
Obviously, the method works reliably only if the inter-individual differences in kinetics are
minimal; for example, the administration protocol must be standardized.

Fitting a function, such as sum of exponentials,
specific model, or performing
spectral analysis, to TACs will provide numerical
parameters that could be compared by simple t-test, ANOVA (Motulsky & Christopoulos, 2004) or
test-retest methods.
Fitting is error-prone and time-consuming process, and it may be difficult to determine how to
weight the different parameters in the comparison.
Selecting appropriate function that would fit all TACs reasonably well without overfitting some TACs
is difficult.
However, Akaike Information Criterium (*AIC*), often used for
model selection, can also be applied in comparing TACs
(Kletting et al., 2009): two equal
functions are fitted to the TACs simultaneously, letting all parameters be freely fitted (local fit),
and constraining a parameter to common value for both TACs (global fit). If, based on *AIC*,
the model with globally fitted parameter value is selected as the best, then equality of the TACs
can be assumed (Kletting et al., 2009).

Fitting can be avoided, if TACs are described with direct parameters such as TAC peak value,
peak time, AUC, area under moment curve (*AUMC*), mean residence
time (*MRT*), or mean transit time (*MTT*).
*AUC* could be calculated from only the initial phase of the PET scan to observe differences
in the perfusion, or in the end phase to represent late distribution.
For the simple TAC comparison, *MRT* could be calculated between measurement start
(*0*) and end time (*T*) as:

## See also:

- Runs test
- Image clustering
- Fitting compartmental models
- Fitting PTACs
- Fitting plasma parent fractions
- Plotting TACs
*AIC*in model selection- Bland-Altman plot
- Simulating noise

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Tags: TAC, Fitting, Validation, Clustering, Filtering, Noise, Statistics

Updated at: 2019-11-22

Created at: 2018-12-14

Written by: Vesa Oikonen