Bayesian Analysis of a One Compartment Kinetic Model Used in Medical Imaging

Mathematical Models for FDG Kinetics in Cancer: A Review

Compartmental analysis is the mathematical framework for the modelling of tracer kinetics in dynamical Positron Emission Tomography. This paper provides a review of how compartmental models are constructed and numerically optimized. Specific focus is given on the identifiability and sensitivity issues and on the impact of complex physiological conditions on the mathematical properties of the models.

PET-ABC: fully Bayesian likelihood-free inference for kinetic models

Aims.We describe an intuitive, easy to use method called PET-ABC that enables full Bayesian statistical inference from single subject dynamic PET data. The performance of PET-ABC was compared with weighted non-linear least squares (WNLS) in terms of reliability of kinetic parameter estimation and statistical power for model selection.Methods.Dynamic PET data based on 1-tissue and 2-tissue compartmental models were simulated with 2 noise models and 3 noise levels. PET-ABC was used to evaluate the reliability of parameter estimates under each condition. It was also used to perform model selection for a simulated noisy dataset composed of a mixture of 1- and 2-tissue compartment kinetics. Finally, PET-ABC was used to analyze a non-steady state dynamic [¹¹C] raclopride study performed on a fully conscious rat administered either 2 mg.kg^-1amphetamine or saline 20 min after tracer injection.Results.PET-ABC yielded posterior point estimates for model parameters with smaller variance than WNLS, as well as probability density functions indicating confidence intervals for those estimates. It successfully identified the superiority of a 2-tissue compartment model to fit the simulated mixed model data. For the drug challenge study, the post observation probability of striatal displacement of the PET signal was 0.9 for amphetamine and approximately 0 for saline, indicating a high probability of amphetamine-induced endogenous dopamine release in the striatum. PET-ABC also demonstrated superior statistical power to WNLS (0.87 versus 0.09) for selecting the correct model in a simulated ligand displacement study.Conclusions.PET-ABC is a simple and intuitive method that provides complete Bayesian statistical analysis of single subject dynamic PET data, including the extent to which model parameter estimates and model choice are supported by the data. Software for PET-ABC is freely available as part of thePETabcpackagehttps://github.com/cgrazian/PETabc.

Validation of Bayesian analysis of compartmental kinetic models in medical imaging

INTRODUCTION

Kinetic compartmental analysis is frequently used to compute physiologically relevant quantitative values from time series of images. In this paper, a new approach based on Bayesian analysis to obtain information about these parameters is presented and validated.

MATERIALS AND METHODS

The closed-form of the posterior distribution of kinetic parameters is derived with a hierarchical prior to model the standard deviation of normally distributed noise. Markov chain Monte Carlo methods are used for numerical estimation of the posterior distribution. Computer simulations of the kinetics of F18-fluorodeoxyglucose (FDG) are used to demonstrate drawing statistical inferences about kinetic parameters and to validate the theory and implementation. Additionally, point estimates of kinetic parameters and covariance of those estimates are determined using the classical non-linear least squares approach.

RESULTS AND DISCUSSION

Posteriors obtained using methods proposed in this work are accurate as no significant deviation from the expected shape of the posterior was found (one-sided P>0.08). It is demonstrated that the results obtained by the standard non-linear least-square methods fail to provide accurate estimation of uncertainty for the same data set (P<0.0001).

CONCLUSIONS

The results of this work validate new methods for a computer simulations of FDG kinetics. Results show that in situations where the classical approach fails in accurate estimation of uncertainty, Bayesian estimation provides an accurate information about the uncertainties in the parameters. Although a particular example of FDG kinetics was used in the paper, the methods can be extended for different pharmaceuticals and imaging modalities.

Sensitivity estimation in time-of-flight list-mode positron emission tomography

PURPOSE

An accurate quantification of the images in positron emission tomography (PET) requires knowing the actual sensitivity at each voxel, which represents the probability that a positron emitted in that voxel is finally detected as a coincidence of two gamma rays in a pair of detectors in the PET scanner. This sensitivity depends on the characteristics of the acquisition, as it is affected by the attenuation of the annihilation gamma rays in the body, and possible variations of the sensitivity of the scanner detectors. In this work, the authors propose a new approach to handle time-of-flight (TOF) list-mode PET data, which allows performing either or both, a self-attenuation correction, and self-normalization correction based on emission data only.

METHODS

The authors derive the theory using a fully Bayesian statistical model of complete data. The authors perform an initial evaluation of algorithms derived from that theory and proposed in this work using numerical 2D list-mode simulations with different TOF resolutions and total number of detected coincidences. Effects of randoms and scatter are not simulated.

RESULTS

The authors found that proposed algorithms successfully correct for unknown attenuation and scanner normalization for simulated 2D list-mode TOF-PET data.

CONCLUSIONS

A new method is presented that can be used for corrections for attenuation and normalization (sensitivity) using TOF list-mode data.