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).

Models should also be tested for the sensitivity to noise in input data (Chen et al., 1991).

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

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).


Software for adding noise to simulated data

Noise model presented above

Gaussian noise without any specific noise model

Measured variation


See also:



References:

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.

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.

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.



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Created at: 2010-09-20
Updated at: 2018-10-17
Written by: Vesa Oikonen