Bayesian non-linear regression with spatial priors for noise reduction and error estimation in quantitative MRI with an application in T1 estimation

Tommy Löfstedt,Anders Garpebring,Mikael Bylund,Max Hellström

doi:10.1088/1361-6560/abb9f5

Abstract

Purpose. To develop a method that can reduce and estimate uncertainty in quantitative MR parameter maps without the need for hand-tuning of any hyperparameters. Methods. We present an estimation method where uncertainties are reduced by incorporating information on spatial correlations between neighbouring voxels. The method is based on a Bayesian hierarchical non-linear regression model, where the parameters of interest are sampled, using Markov chain Monte Carlo (MCMC), from a high-dimensional posterior distribution with a spatial prior. The degree to which the prior affects the model is determined by an automatic hyperparameter search using an information criterion and is, therefore, free from manual user-dependent tuning. The samples obtained further provide a convenient means to obtain uncertainties in both voxels and regions. The developed method was evaluated on T 1 estimations based on the variable flip angle method. Results. The proposed method delivers noise-reduced T 1 parameter maps with associated error estimates by combining MCMC sampling, the widely applicable information criterion, and total variation-based denoising. The proposed method results in an overall decrease in estimation error when compared to conventional voxel-wise maximum likelihood estimation. However, this comes with an increased bias in some regions, predominately at tissue interfaces, as well as an increase in computational time. Conclusions. This study provides a method that generates more precise estimates compared to the conventional method, without incorporating user subjectivity, and with the added benefit of uncertainty estimation.

Highlights

Quantitative magnetic resonance imaging can provide measurements of tissue properties that are independent of the exact details of the data acquisition, and simultaneously often provide interpretations of measurements (Tofts 2003)
To determine the value of the hyperparameter κ, the only unknown hyperparameter in this work, we propose to use the widely applicable information criterion (WAIC, known as the Watanabe–Akaike information criterion) (Watanabe 2009)
Tissue parameter estimation Parameter maps computed using all three estimators are shown in figure 2

Summary

Introduction

Quantitative magnetic resonance imaging (qMRI) can provide measurements of tissue properties that are independent of the exact details of the data acquisition, and simultaneously often provide interpretations of measurements (Tofts 2003). Several applications of qMRI can be found in cancer diagnostics and follow-ups, e.g. prostate cancer staging using apparent diffusion coefficient imaging (Fütterer 2016) and dynamic contrast-enhanced MRI (DCE-MRI) for early response assessment (Pham et al 2017). Structured spatial regularisation terms and priors have been used in qMRI by several authors They have, for instance, been used when estimating relaxation times and proton density (Wang and Cao 2012, Baselice et al 2016, Kumar et al 2012, Raj et al 2014), in diffusion and intra-voxel incoherent motion (IVIM) estimation (While 2017, Orton et al 2014), for B0-estimation (Baselice et al 2010), and for dynamic contrast-enhanced MRI (DCE-MRI) (Schmid et al 2006, Kelm et al 2009, Sommer and Schmid 2014, Bartos et al 2019)

Objectives

Results

Conclusion