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.

Fitting a function, such as sum of exponentials, or model 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:


Bradley JV. Distribution-Free Statistical Tests. Wright Air Development Division, 1960.

Choudhary PK, Nagaraja HN: Measuring Agreement: Models, Methods, and Applications. Wiley, 2018. ISBN 978-1-118-07858-7.

Cobelli C, Forster D, Toffolo G: (2002) Tracer Kinetics in Biomedical Research: From Data to Model. Kluwer Academic Publishers.

Grant DA. Additional tables of the probability of runs of correct responses in learning and problem-solving. Psychol Bull. 1947; 44(3): 276-279. doi: 10.1037/h0054957.

Herholz K, Heiss WD, Pietrzyk U, Wienhard K. Pixel-by-pixel fits of blood volume, transport, and metabolic processes: principle, normal values, and brain tumor studies with dynamic FDG-PET. In: Beckers C, Goffinet A, Bol A (eds.): Positron Emission Tomography in Clinical Research and Clinical Diagnosis: Tracer Modelling and Radioreceptors, pp 148-161. Kluwer, 1989. ISBN: 0-7923-0254-0.

Kletting P, Kull T, Reske SN, Glatting G: Comparing time activity curves using the Akaike information criterion. Phys Med Biol. 54: N501-N507, 2009b. doi: 10.1088/0031-9155/54/21/N01

Motulsky H, Christopoulos A. 2004. Fitting Models to Biological Data Using Linear and Nonlinear Regression. Oxford University Press, NY.

Motulsky HJ, Ransnas LA. Fitting curves to data using nonlinear regression: a practical and nonmathematical review. FASEB J. 1987; 1: 365-374. doi: 10.1096/fasebj.1.5.3315805.

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.

NIST/SEMATECH e-Handbook of Statistical Methods,

Wackerly D, Mendenhall W, Scheaffer RL. Mathematical Statistics with Applications, 7th ed., Cengage Learning, 2008. ISBN-13: 978-0495110811.

Young P: Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask. Springer, 2005. ISBN 978-3-319-19051-8.

Tags: , , , , , ,

Created at: 2018-12-14
Updated at: 2018-12-17
Written by: Vesa Oikonen