Simulation of noise in PET data
There is no generally agreed method to quantify statistical noise in PET images. Statistics of nuclear decay follow the binomial law, which, in the level of event detection with PET detectors can be well approximated with Poisson distribution (Sitek and Celler, 2015). Yet, Poisson distribution is not adequate for reconstructed PET images (Budinger et al., 1978). Since several additive sources of error tend to form a Gaussian distribution, Gaussian noise model is often applied to PET data (Fessler, 1994). More recently, Conwell-Maxwell-Poisson (CMP) noise model (Santarelli et al., 2016) and negative binomial (NM) distribution model have been proposed (Santarelli et al., 2017). Noise equivalent count (NEC) rates and signal-to-noise ratio (S/N, SNR) have a linear relationship (Dahlbom et al., 2005). Variance and other statistical properties of PET images can be estimated using bootstrap method (Dahlbom et al., 2002).
Inside the image, noise distribution can be assumed uniform even when the radioactivity concentration is heterogeneous (Asselin et al., 2004). The method used for global image noise measurements in CT scans can be applied to PET SUV images (Sartoretti et al., 2023).
Models should also be tested for the sensitivity to noise in input data (Huesman & Mazoyer, 1987; Chen et al., 1991). Tissue data is simulated with noiseless input data, followed by analysis with the same input data but with added noise, and possibly with other errors such as dispersion and delay. Note that noise model for manually drawn plasma data should not be based on the radioactivity concentration, if samples are measured using count-limit.
One option to add noise to simulated tissue time-activity curves is to use empirical noise, assessed as the deviations of measured and fitted curves from actual PET data analysis (Huang et al., 2018).
In data analysis, the variable noise level in time frames can be accounted for by weighting data points during model fitting. Widely used weighting methods are based on the estimated measurement variance, like the noise model (Mazoyer et al., 1986; Jovkar et al., 1989; Chen et al., 1991; Logan et al., 2001; Varga & Szabo, 2002). Assuming Poisson distribution, error of measured counts is assumed to equal the square root of measured counts (events)
, and the counts measured during time frame Δt can be calculated from the decay corrected and calibrated radioactivity concentration (C)
, where exponential term is used to remove the decay correction, and proportionality coefficient PC removes the calibration and other corrections applied to the image. C is the average concentration in the whole image during the time frame. Coefficient of variation is the same for radioactivity concentrations and counts:
, and, continuing with the counts
Noise can be added to simulated noiseless concentration Csim with equation
, where G(0,1) is randomly generated number of Gaussian distribution with zero mean and SD of 1 (Logan et al., 2001; Varga & Szabo, 2002). In place of Gaussian distribution, uniform variance or white noise has been used in some instances (Coxson et al., 1991). Notice that PC is in some publications placed outside of the square root.
In simulated data the early time frames the concentration can be zero, leading to no noise when using these equations, while in real PET studies some noise is observed in these zero activity time frames, too. For noise simulations the SD can be set to a predetermined value for these frames (Li et al., 2022), or a minimum CV can be set for every time frame.
Software for adding noise to simulated data
Noise model presented above
- Add Gaussian noise to simulated TACs
- Add Gaussian noise to simulated PET images
- Compute SD and CV predicted by noise model
Gaussian noise without any specific noise model
- For simulated TACs, before time frames are simulated, noise with specified CV% and/or SD can be added using programs svar4tac and var4tac
- Noise with specified CV%, depending on weights, can be added with wvar4dat.
Multiple noise realizations
In Monte Carlo simulations numerous copies of the same TAC must be made, each with different
noise set added. Program svar4tac with option
-x can be used to make the copies, either with noise or without (by setting CV to zero).
Program fvar4tac can optionally (
a number of noise realizations of the same data set in separate files.
- Analysis of simulated data
- Simulation of PET time frames
- Simulation of PET image
- Models for simulation
- Tissue TAC data
- Quantification of radioactivity in PET studies
- Weights in analysis
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Updated at: 2023-09-18
Created at: 2010-09-20
Written by: Vesa Oikonen