Correlation coefficient constrained parametric images

Multiple-time graphical analyses are fast and generally robust enough methods to be applied to dynamic PET images, producing parametric images of distribution volume, V_T, using Logan plot, or net influx rate, K_i, using Patlak plot. There are numerous tools to calculate parametric V_T and K_i images from dynamic PET images; Open Source programs imgdv and imgki, with command-line interface, have been developed in Turku PET Centre for these purposes.

Individual time frames in the dynamic image are usually very noisy, and can include large negative pixel values because of scatter correction and image reconstruction artefacts. Especially FBP reconstruction method produces positive/negative streak artefacts that affect not only the background in the image but penetrate also the areas of interest. Pixel-by-pixel calculation of Patlak and Logan plot from the noisy pixel values leads to noisy K_i and V_T images, and even though the noise level is lower than in the individual time frame images, it usually is higher than in late-scan SUV images based on a long time frame. In Logan plot, noisy data leads to biased V_T (Logan, 2003). In diagnostic studies the noise-induced image artefacts complicate detection of lesions. One solution to this problem is to generate correlation coefficient constrained Patlak parametric images (r-constrained images) from dynamic PET data by zeroing the pixel K_i if the correlation coefficient (r) of the Patlak plot is below a threshold value (Zasadny & Wahl, 1993 and 1996). The appropriate level of correlation coefficient filtering must be empirically determined. This procedure results in marked qualitative improvement in [¹⁸F]FDG K_i images (Zasadny & Wahl, 1996). This technique can be adapted in defining the VOIs and calculation of lesion volume and metabolic index (Wu et al., 1996).

In Patlak plot, data is transformed and plotted in a x_i,y_i graph, and line is fitted to the linear phase of the plot, omitting the non-linear phase in the beginning. The slope of the fitted line represents the K_i of the radioligand. The weighted correlation coefficient (r) can be calculated from equation:

${r} = \tfrac{\sum_{i=1}^{N} {w_i} \sum_{i=1}^{N} {w_i}{x_i}{y_i} - \ \sum_{i=1}^{N} {w_i}{x_i} \sum_{i=1}^{N} {w_i}{y_i} }{ \ \sqrt{ (\sum_{i=1}^{N} {w_i} \sum_{i=1}^{N} {w_i}{x_i^2} - (\sum_{i=1}^{N} {w_i}{x_i})^2) \ (\sum_{i=1}^{N} {w_i} \sum_{i=1}^{N} {w_i}{y_i^2} - (\sum_{i=1}^{N} {w_i}{y_i})^2) }}$

, where i=1 is the first time frame and N is the number of time frames included in the line fit, and w_i is the weight for each individual time frame. Zasadny & Wahl (1996) reported that weights were calculated from image pixel concentrations as:

${w_i} = \frac{1}{C_{T}(t_i)}$

However, variance estimation should not be based on individual image pixel values, but on the number of events collected during the time frame (Mazoyer et al., 1986). Indeed, in later studies the weights have been based on sinogram total events, c_i, and time frame lengths, Δt_i (Karakatsanis et al., 2013 and 2015):

${w_i} = \frac{(\Delta t_i)^2}{c_i}$

Since there is no generally agreed method to quantify statistical noise in PET images, many alternative weighting schemes have been proposed.

References

Karakatsanis NA, Lodge MA, Zhou Y, Wahl RL, Rahmim A. Dynamic whole-body PET parametric imaging: II. Task-oriented statistical estimation. Phys Med Biol. 2013; 58: 7419-7445. doi: 10.1088/0031-9155/58/20/7419.

Karakatsanis NA, Zhou Y, Lodge MA, Casey ME, Wahl RL, Zaidi H, Rahmim A. Generalized whole-body Patlak parametric imaging for enhanced quantification in clinical PET. Phys Med Biol. 2015; 60: 8643-8673. doi: 10.1088/0031-9155/60/22/8643.

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.

Logan J. A review of graphical methods for tracer studies and strategies to reduce bias. Nucl Med Biol. 2003; 30: 833–844. doi: 10.1016/S0969-8051(03)00114-8.

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.

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.

Zasadny KR, Wahl RL. Fit-constrained ("tumor-tailored") parametric images of cancer using FDG. J Nucl Med. 1993; 34: 41P.

Zasadny KR, Wahl RL. Enhanced FDG-PET tumor imaging with correlation-coefficient filtered influx-constant images. J Nucl Med. 1996; 37: 371-374. PMID: 8667078.

Tags: Filtering, Noise

Updated at: 2019-08-08
Created at: 2018-10-14
Written by: Vesa Oikonen

Correlation coefficient constrained parametric images

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References