Estimation of Noise-Free Variance to Measure Heterogeneity

Variance is a statistical parameter used to characterize heterogeneity or variability in data sets. However, measurements commonly include noise, as random errors superimposed to the actual value, which may substantially increase the variance compared to a noise-free data set. Our aim was to develop and validate a method to estimate noise-free spatial heterogeneity of pulmonary perfusion using dynamic positron emission tomography (PET) scans. On theoretical grounds, we demonstrate a linear relationship between the total variance of a data set derived from averages of n multiple measurements, and the reciprocal of n. Using multiple measurements with varying n yields estimates of the linear relationship including the noise-free variance as the constant parameter. In PET images, n is proportional to the number of registered decay events, and the variance of the image is typically normalized by the square of its mean value yielding a coefficient of variation squared (CV²). The method was evaluated with a Jaszczak phantom as reference spatial heterogeneity (CV_r²) for comparison with our estimate of noise-free or ‘true’ heterogeneity (CV_t²). We found that CV_t² was only 5.4% higher than CV_r². Additional evaluations were conducted on 38 PET scans of pulmonary perfusion using ¹³NN-saline injection. The mean CV_t² was 0.10 (range: 0.03–0.30), while the mean CV² including noise was 0.24 (range: 0.10–0.59). CV_t² was in average 41.5% of the CV² measured including noise (range: 17.8–71.2%). The reproducibility of CV_t² was evaluated using three repeated PET scans from five subjects. Individual CV_t² were within 16% of each subject''s mean and paired t-tests revealed no difference among the results from the three consecutive PET scans. In conclusion, our method provides reliable noise-free estimates of CV_t² in PET scans, and may be useful for similar statistical problems in experimental data.