$μ$GUIDE: a framework for quantitative imaging via generalized uncertainty-driven inference using deep learning

Abstract

This work proposes $μ$GUIDE: a general Bayesian framework to estimate posterior distributions of tissue microstructure parameters from any given biophysical model or MRI signal representation, with exemplar demonstration in diffusion-weighted MRI. Harnessing a new deep learning architecture for automatic signal feature selection combined with simulation-based inference and efficient sampling of the posterior distributions, $μ$GUIDE bypasses the high computational and time cost of conventional Bayesian approaches and does not rely on acquisition constraints to define model-specific summary statistics. The obtained posterior distributions allow to highlight degeneracies present in the model definition and quantify the uncertainty and ambiguity of the estimated parameters.

Figures (14)

• Figure 1: µGUIDE framework. µGUIDE takes as input an observed data vector and relies on the definition of a biophysical or computational model ascoli2007neuromorphocallaghan_config_2020jelescu_challenges_2020. It outputs a posterior distribution of the model parameters. Based on a SBI framework, it combines a Multi-Layer Perceptron (MLP) with 3 layers and a Neural Posterior Estimator (NPE). The MLP learns a low-dimensional representation of $\boldsymbol{x}$, based on a small number of features ($N_f$), that can be either defined a priori or determined empirically during training. The MLP is trained simultaneously with the NPE, leading to the extraction of the optimal features that minimize the bias and uncertainty of $p(\boldsymbol{\theta} | \boldsymbol{x})$.
• Figure 1: Number of degenerate cases per parameter on 10000 noise-free simulations. Training and estimations of the posterior distributions was performed on CPU. Time for training each model and time for estimating posterior distributions of 10000 noise-free simulations, define if they are degenerate or not, and extract the MAP, uncertainty, ambiguity are also reported.
• Figure 2: µGUIDE summarizes information contained in the estimated posterior distributions. A) Examples of degenerate and non-degenerate posterior distributions. Two Gaussian distributions are fitted to the obtained posterior distribution, where the means and standard deviations are represented by the vertical lines and shaded areas. A voxel is considered as degenerate if the derivative of the fitted Gaussian distributions changes signs more than once (i.e. multiple local maxima), and if the two Gaussian distributions are not overlapping (the distance between the two Gaussian means is inferior to the sum of their standard deviations). B) Presentation of the measures introduced to quantify a posterior distribution on exemplar non-degenerate posterior distributions. Maximum A Posteriori (MAP): is the most likely parameter estimate (dashed vertical lines). Uncertainty: measures the dispersion of the 50% most probable samples using the interquartile range, with respect to the prior range. Ambiguity: measures the Full Width at Half Maximum (FWHM), in percentage with respect to the prior range.
• Figure 2: Number of degenerate cases per parameter on 10000 noisy simulations (Rician noise with $\text{SNR}=50$). Training and estimations of the posterior distributions was performed using a GPU. Time for training each model and time for estimating posterior distributions of 10000 noisy simulations, define if they are degenerate or not, and extract the MAP, uncertainty, ambiguity are also reported.
• Figure 3: Comparison between µGUIDE and MCMC. A) Posterior distributions obtained using either µGUIDE or MCMC on three exemplar simulations with model 2 (SM - $\text{SNR}=50$). Names of the model parameters are indicated in the titles of the panels. B) Bias between the ground truth values used for simulating the diffusion signals, and the maximum-a-posteriori extracted from the posterior distributions using either µGUIDE or MCMC. Sharper and less biased posterior distributions are obtained using µGUIDE.