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

Abstract

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.

Machine learning based priors for Bayesian inversion in MR imaging

“…Inverse problems are frequently encountered in metrology. Examples include applications such as scatterometry [1], magnetic resonance imaging (MRI) [2], magnetic resonance fingerprinting [3,4], cardiac perfusion [5], computed tomography [6], and spectroscopy [7], to mention just a few. Statistical modeling is an appropriate tool to tackle inverse problems and provides the basis for established procedures such as maximum likelihood estimation from classical statistics [8] or the application of Bayesian inference [9].…”

“…It is also common to include expected properties of the solution. For example, spatial smoothness can be expressed by a Gaussian Markov random field (GMRF) [18] taken as a prior distribution (see applications such as cardiac perfusion [5], nano-spectroscopy [19] and magnetic resonance fingerprinting [11]). Another thriving field of research deals with the determination of prior knowledge from data.…”

Parallelized integrated nested Laplace approximations for fast Bayesian inference

“…Bayesian approaches to spatial-temporal modeling have shown to be a versatile tool to capture complex phenomena and handle imbalanced or missing observations [1,2]. Among their wide range of applications are climate and weather modeling [3], disease mapping [4,5], medical image analysis [6,7], traffic management [8,9], environmental changes [10,11,12] and health economic evaluations [13]. We see growing amounts of data available to describe these phenomena and increasingly sophisticated models to characterize them [14,15].…”

Classification of Myocardial Blood Flow based on Dynamic Contrast-Enhanced Magnetic Resonance Imaging using Hierarchical Bayesian Models

“…We note that the application of HBMs to myocardial perfusion modelling has been proposed in four previous publications: Schmid et al (2007), Schmid (2011), Lehnert et al (2019) and Scannell et al (2020). Here, the emphasis has been on the estimation of the MBF per se, and the authors have demonstrated that the MBF parameters can be more accurately estimated than with traditional methods, such as non-hierarchical non-linear regression, particularly in regimes where the signal-to-noise ratio (SNR) is low (see Scannell et al, 2020 for details).…”

“…Schmid (2011) extended the modelling in Schmid et al (2007) by importing spatio-temporal constrains to the B-spline model using Gaussian MRF. Lehnert et al (2019) illustrated an HBM method to analyse the myocardial perfusion DCE-MRI. In this method, a large-scale Bayesian spatio-temporal regression method was applied to model the data.…”