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)

${\Delta c} = \sqrt{c}$

, and the counts measured during time frame Δt can be calculated from the decay corrected and calibrated radioactivity concentration (C)

$c = \frac{C \times \Delta t \times e^{-\lambda t}}{P_C}$

, where exponential term is used to remove the decay correction, and proportionality coefficient P_C 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:

${CV} = \frac{\Delta C}{C} = \frac{\Delta c}{c}$

, and, continuing with the counts

${CV} = \sqrt{\frac{1}{c}}$

Thus,

${CV} = \sqrt{\frac{P_C}{C \times \Delta t \times e^{-\lambda t}}}$

Noise can be added to simulated noiseless concentration C_sim with equation

${C_{noisy}} = {C_{sim}} \times (1 + {CV}{\times}{G(0,1)})$

, 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 P_C 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

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.

Measured variation

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 (-R) make a number of noise realizations of the same data set in separate files.

Literature

Asselin M-C, Cunningham VJ, Amano S, Gunn RN, Nahmias C. Parametrically defined cerebral blood vessels as non-invasive blood input functions for brain PET studies. Phys Med Biol. 2004; 49: 1033-1054. doi: 10.1088/0031-9155/49/6/013.

Budinger TF, Derenzo SE, Gullberg GT, Greenberg WL, Huesman RH. Emission computer assisted tomography with single-photon and positron annihilation photon emitters. J Comput Assist Tomogr. 1977; 1:131-145. doi: 10.1097/00004728-197701000-00015.

Budinger TF, Derenzo SE, Greenberg WL, Gullberg GT, Huesman RH. Quantitative potentials of dynamic emission computed tomography. J Nucl Med. 1978; 19(3): 309-315. PMID: 632910.

Chen K, Huang SC, Yu DC. The effects of measurement errors in the plasma radioactivity curve on parameter estimation in positron emission tomography. Phys Med Biol. 1991; 36:1183-1200. doi: 10.1088/0031-9155/36/9/003.

Coxson PG, Huesman RH, Borland L. Consequences of using a simplified kinetic model for dynamic PET data. J Nucl Med. 1997; 38:660-667. PMID: 9098221.

Huesman RH, Mazoyer BM. Kinetic data analysis with a noisy input function. Phys Med Biol. 1987; 32(12): 1569-1579. doi: 10.1088/0031-9155/32/12/004.

Logan J, Fowler JS, Volkow ND, Ding YS, Wang GJ, Alexoff DL. A strategy for removing the bias in the graphical analysis method. J Cereb Blood Flow Metab. 2001; 21(3): 307-320. doi: 10.1097/00004647-200103000-00014.

Santarelli MF, Della Latta D, Scipioni M, Positano V, Landini L. A Conway-Maxwell-Poisson (CMP) model to address data dispersion on positron emission tomography. Comput Biol Med. 2016; 77: 90-101. doi: 10.1016/j.compbiomed.2016.08.006.

Varga J, Szabo Z. Modified regression model for the Logan plot. J Cereb Blood Flow Metab. 2002; 22(2): 240-244. doi: 10.1097/00004647-200202000-00012.

Walker MD, Matthews JC, Asselin M-C, Watson CC, Saleem A, Dickinson C, Charnley N, Julyan PJ, Price PM, Jones T. Development and validation of a variance model for dynamic PET: uses in fitting kinetic data and optimizing the injected activity. Phys Med Biol. 2010; 55: 6655-6672. doi: 10.1088/0031-9155/55/22/005.

Yaqub M, Boellaard R, Kropholler MA, Lammertsma AA. Optimization algorithms and weighting factors for analysis of dynamic PET studies. Phys Med Biol. 2006; 51: 4217-4232. doi: 10.1088/0031-9155/51/17/007.

Tags: Simulation, Noise, Variance, Bootstrap, Filtering, Weighting, Statistics

Updated at: 2023-09-18
Created at: 2010-09-20
Written by: Vesa Oikonen