Large-Scale Bayesian Spatial-Temporal Regression with Application to Cardiac MR-Perfusion Imaging

We develop a hierarchical Bayesian approach for the inference of large-scale spatial-temporal regression as often encountered in the analysis of imaging data. For each spatial location a linear temporal Gaussian regression model is considered. Large-scale refers to the large number of spatially distributed regression parameters to be inferred. The spatial distribution of the sought regression parameters, which typically represent physical quantities, is assumed to be smooth and bounded from below. Truncated, intrinsic Gaussian Markov random field priors are employed to express this prior knowledge. The dimensionality of the spatially distributed parameters is high, which challenges the calculation of results using standard Markov chain Monte Carlo procedures. A calculation scheme is developed that utilizes an approximate analytical expression for the marginal posterior of the amount of smoothness and the strength of the noise in the data, in conjunction with a conditional high-dimensional truncated Gaussian distribution for the spatial distribution of the regression parameters. We prove propriety of the posterior and explore the existence of its moments. The proposed approach is applicable to high-dimensional imaging problems arising in the use of Gaussian Markov random field priors for inferring spatially distributed parameters from spatial or spatial-temporal data. We demonstrate its application for the quantification of myocardial perfusion imaging from first-pass contrast-enhanced cardiovascular magnetic resonance data. The Bayesian approach allows for a quantification of perfusion, including a complete characterization of uncertainties which is desirable in diagnostics. It therefore provides not just a perfusion estimate but also a measure for how reliable this estimate is.