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Bayesian kinetic modelling
Poster Presentations / NeuroImage 31 (2006) T44 – T186
T71
Poster Presentation No.: 027
Bayesian kinetic modelling Roger Gunn, V. Schmid, B. Whitcher, V. Cunningham GSK, Greenford, UK
The estimation of appropriate neuroreceptor parameters from dynamic PET data using a suitable compartmental structure and input function is the cornerstone of quantification for all neuroreceptor studies. This is normally achieved in a weighted least squares framework which is equivalent to the maximum likelihood estimate. Such estimation provides point estimates of the parameter values and allows for standard errors to be determined by asymptotic statistics. Here, we introduce the approach of Bayesian estimation for the quantification of dynamic neuroreceptor PET data. The Bayesian framework allows the introduction of prior information (by specifying distributions for the kinetic parameters) and returns a posterior distribution characterising how likely different parameter values are given the observed data and the prior information. The method is founded on Bayes rule and can be expressed as P(a)y) = P(y)a) P(a)/P(y) where y is the data and a is the parameters. This expresses the conditional distribution of the parameters P(a)y), after observing the data (posterior distribution) as a function of the likelihood, P(y)a) the prior P(a) and the evidence P(y). In practice, the evidence function can be dropped from calculations as it is a constant. Other than for simple likelihood and prior distributions, it is necessary to employ iterative sampling methods to determine the posterior distribution. We have implemented the current Bayesian estimation techniques using the Metropolis – Hastings algorithm which is a Markov Chain Monte Carlo (MCMC) method. This has the advantage that we are not restricted in terms of the forms for the likelihood and prior distributions that we can use. Once one has obtained the posterior distribution, one can calculate a number of estimates from it. If we are looking for a single value for each of the parameters, then the most probable value, the maximum a posteriori (MAP) estimate, is given by maximising the posterior. In addition, other values which characterise the accuracy of the parameter estimates can be obtained from the posterior distribution. We demonstrate the Bayesian approach to kinetic modelling by applying it to standard compartmental models used in the analysis of dynamic PET data. Particular examples are drawn from ongoing imaging studies being employed in drug development. Reference: Box, G., Tiao, G., 1973. Bayesian inference in Statistical Analysis. In: Wiley Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (eds.) (1996). Markov Chain Monte Carlo in Practice. Chapman and Hall. doi:10.1016/j.neuroimage.2006.04.061
T71
Poster Presentation No.: 027
Bayesian kinetic modelling Roger Gunn, V. Schmid, B. Whitcher, V. Cunningham GSK, Greenford, UK
The estimation of appropriate neuroreceptor parameters from dynamic PET data using a suitable compartmental structure and input function is the cornerstone of quantification for all neuroreceptor studies. This is normally achieved in a weighted least squares framework which is equivalent to the maximum likelihood estimate. Such estimation provides point estimates of the parameter values and allows for standard errors to be determined by asymptotic statistics. Here, we introduce the approach of Bayesian estimation for the quantification of dynamic neuroreceptor PET data. The Bayesian framework allows the introduction of prior information (by specifying distributions for the kinetic parameters) and returns a posterior distribution characterising how likely different parameter values are given the observed data and the prior information. The method is founded on Bayes rule and can be expressed as P(a)y) = P(y)a) P(a)/P(y) where y is the data and a is the parameters. This expresses the conditional distribution of the parameters P(a)y), after observing the data (posterior distribution) as a function of the likelihood, P(y)a) the prior P(a) and the evidence P(y). In practice, the evidence function can be dropped from calculations as it is a constant. Other than for simple likelihood and prior distributions, it is necessary to employ iterative sampling methods to determine the posterior distribution. We have implemented the current Bayesian estimation techniques using the Metropolis – Hastings algorithm which is a Markov Chain Monte Carlo (MCMC) method. This has the advantage that we are not restricted in terms of the forms for the likelihood and prior distributions that we can use. Once one has obtained the posterior distribution, one can calculate a number of estimates from it. If we are looking for a single value for each of the parameters, then the most probable value, the maximum a posteriori (MAP) estimate, is given by maximising the posterior. In addition, other values which characterise the accuracy of the parameter estimates can be obtained from the posterior distribution. We demonstrate the Bayesian approach to kinetic modelling by applying it to standard compartmental models used in the analysis of dynamic PET data. Particular examples are drawn from ongoing imaging studies being employed in drug development. Reference: Box, G., Tiao, G., 1973. Bayesian inference in Statistical Analysis. In: Wiley Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (eds.) (1996). Markov Chain Monte Carlo in Practice. Chapman and Hall. doi:10.1016/j.neuroimage.2006.04.061