Covariance expressions for implicit estimators of kinetic parameters with dynamic PET

Kinetic parameter estimation with dynamic PET is used in several applications including but not limited to tumor and diabetes detection, neuroscience, and drug delivery. We can estimate the exchange rate constants of radioactive tracer between a region of interest and the blood using the time-activity-curves (TACs) acquired with PET. Although several methods have been developed for estimation, there has not been much study on the statistical properties such as the covariance of these estimators. Current studies are either based on time consuming numerical methods, or use the inverse of the Fisher information matrix as an asymptotic approximation to the covariance matrix. The covariance of the kinetic parameters is an important quantity that determines the reliability of the estimates. Since there is no closed-form for the estimates of the kinetic parameters from the measured TAC, we use the implicit function theorem and derive expressions for the covariance of these parameters. The results are verified by Monte Carlo simulations. The expressions derived are useful for example determining an acceptable minimum SNR value for a certain application, which directly has an effect on the count number and hence the dose level. Using our results, it will be possible to setup the dose level based on a desired accuracy of the kinetic parameters to be estimated.