# 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. More recently, Conwell-Maxwell-Poisson (CMP) noise model has been proposed (Santarelli et al., 2016). Noise equivalent count (NEC) rates and signal-to-noise ratio (S/N) have a linear relationship (Dahlbom et al., 2005). Variance and other statistical properties of PET images can be estimated using bootstrap method.

Inside the image, noise distribution can be assumed uniform even when the radioactivity concentration is heterogeneous (Asselin et al., 2004).

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. Coefficient of variation is the same for radioactivity concentrations and counts:

, and, continuing with the counts

Thus,

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). Notice that PC is in some publications placed outside of the square root. In place of Gaussian distribution, uniform variance or white noise has been used in some instances (Coxson et al., 1991).

## References:

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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.

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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.

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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.

Updated at: 2018-12-29
Created at: 2010-09-20
Written by: Vesa Oikonen