Multiparameter Uncertainty Mapping in Quantitative Molecular MRI using a Physics-Structured Variational Autoencoder (PS-VAE)

TL;DR

A physics-structured variational autoencoder designed for rapid extraction of voxelwise multi-parameter posterior distributions that is in good agreement with those calculated using a brute-force Bayesian analysis, while providing an orders-of-magnitude acceleration in whole brain quantification.

Abstract

Quantitative imaging methods, such as magnetic resonance fingerprinting (MRF), aim to extract interpretable pathology biomarkers by estimating biophysical tissue parameters from signal evolutions. However, the pattern-matching algorithms or neural networks used in such inverse problems often lack principled uncertainty quantification, which limits the trustworthiness and transparency, required for clinical acceptance. Here, we describe a physics-structured variational autoencoder (PS-VAE) designed for rapid extraction of voxelwise multi-parameter posterior distributions. Our approach integrates a differentiable spin physics simulator with self-supervised learning, and provides a full covariance that captures the inter-parameter correlations of the latent biophysical space. The method was validated in a multi-proton pool chemical exchange saturation transfer (CEST) and semisolid magnetization transfer (MT) molecular MRF study, across in-vitro phantoms, tumor-bearing mice, healthy human volunteers, and a subject with glioblastoma. The resulting multi-parametric posteriors are in good agreement with those calculated using a brute-force Bayesian analysis, while providing an orders-of-magnitude acceleration in whole brain quantification. In addition, we demonstrate how monitoring the multi-parameter posterior dynamics across progressively acquired signals provides practical insights for protocol optimization and may facilitate real-time adaptive acquisition.

Figures (8)

• Figure 1: Computational pipelines for multi-parameter posterior estimation in quantitative molecular MRI. (a) Rapid uncertainty estimation using a physics-structured variational autoencoder (PS-VAE). A multi-layer perception ($\mathcal{E}_w$, red) is trained using self-supervised learning on experimental signals $S_{exp} \in \mathcal{S}_{train}$. At inference, voxelwise parameterization $\hat{\phi}({S_{exp}},w)=\hat{\mu},\hat{\boldsymbol{\Sigma}}=\mathcal{E}_w(S_m)$ of an approximate posterior $\hat{P}(\theta|S_{exp}) \approx Q_{\hat{\phi}} = \mathcal{N}(\hat{\boldsymbol{\mu}}, \hat{ \boldsymbol{\Sigma} })$ is obtained. The derived confidence region (CR) $\Theta_{CR} \subset \Theta$ in the latent space of the tissue parameters $\Theta$, reflects consistency with the measured signal under a biophysical model ($\mathcal{F}$, blue), so that $\mathcal{F}(\theta' \in \Theta_{CR}) \approx S_{exp}$. The CR is an oriented uncertainty ellipsoid around the mean estimate $\hat{\boldsymbol{\mu}}$, represented using a non-diagonal covariance matrix $\hat{\boldsymbol{\Sigma}}$. A random posterior sample $\theta' \sim Q_{\hat{\phi}}$ is decoded at each iteration during training, and a $\boldsymbol{\Sigma}$-regularizing term balances the cycle-consistency loss. (b) Reference framework for likelihood-mapping. Each experimental signal $S_{exp}$, associated with a given voxel, is compared (by means of dot-product) to a local dictionary of signals across a dense grid in the parameter space $\Theta =\Theta_1 \times \Theta_2$. The discrepancy map is then translated into a likelihood and, consequently, into the posterior.
• Figure 2: Uncertainty-aware quantification of the semisolid MT (a-d, i-l) and amide (e-h, m-p) proton exchange rates (top) and volume fractions (bottom) in tumor-bearing mice. The color-coded brain images (a,c,e,g,i,k,m,o) show the PS-VAE output maps of maximum-a-posteriori (MAP) estimates of tissue parameters (center) and the corresponding univariate confidence interval (CI) bounds; the distributions across the entire slice are displayed inside the colorbar. Multi-parametric confidence regions (CR) are provided alongside each parameter map (b,d,f,h,j,l,n,p) for randomly chosen pixels in the ipsitumoral and contralateral regions of interest (ROIs), marked by "A" and "B" within the MAP images. The CRs were obtained from Gaussian posteriors using PS-VAE output (cyan) or from reference grid-based method (free-form CR bounds in magenta, underlying posteriors as heatmaps). The agreement between the two posteriors as quantified using the Mahalanobis distance is provided as text within the figure.
• Figure 3: Uncertainty-aware quantification of the semisolid MT proton exchange rates (a,c,e,g, top) and volume fractions (a,c,e,g, bottom) in four healthy human volunteers. The color-coded brain images show the PS-VAE output maps of maximum-a-posteriori (MAP) estimates of tissue parameters (center) and the corresponding univariate confidence interval (CI) bounds; the distributions across the entire slice are displayed inside the colorbar. Multi-parametric confidence regions (CR) are provided alongside each output parameter map (b,d,f,h) for randomly chosen pixels in the white and gray matter regions of interest (ROIs), as marked by "A" and "B" within the MAP images. The CRs were obtained from either Gaussian posteriors using the PS-VAE output (cyan) or a reference grid-based method (free-form CR bounds in magenta, underlying posteriors as heatmaps). The agreement between the two posteriors as quantified using the Mahalanobis distance is provided as text within the figure. The per-subject PS-VAE maps were obtained using a network trained on all other subjects (LOOCV, as described in section \ref{['sec:loocv-transfer-eval']}).
• Figure 4: Uncertainty-aware quantification of the semisolid MT proton exchange rates (a, top) and volume fractions (a, bottom) in a patient with glioblastoma. The color-coded brain images represent the PS-VAE output maps of maximum-a-posteriori (MAP) estimates of tissue parameters (center) and the corresponding univariate confidence interval (CI) bounds; the distributions across the entire slice are displayed inside the colorbar. Multi-parametric confidence regions (CR) are provided in (b,c) for randomly chosen pixels in the ipsitumoral and contralateral regions of interest (ROIs), marked by 'A','B','C', and 'D' within the MAP images. The CRs were obtained from Gaussian posteriors using the PS-VAE output (cyan) or from a reference grid-based method (free-form CR bounds in magenta, underlying posteriors as heatmaps). The agreement between the two posteriors as quantified using the Mahalanobis distance is provided as text within the figure.
• Figure 5: Quantitative performance evaluation and statistical overview of the PS-VAE-derived output with respect to the reference method.(a-d). Correlation plots comparing the NRMSE (reflecting the goodness of fit) of the MAP parameter values obtained by PS-VAE (y-axis) and the reference exact Bayesian quantification method (x-axis), for all voxels of quantification of semisolid MT in mice (a), APT in mice (b), semisolid MT in a patient with glioblastoma (c), and in healthy volunteers (d). (e). NRMSE maps for PS-VAE-derived MAP values across four healthy volunteers. (f-i). Complete distribution of the Mahalanobis distances of samples from PS-VAE-derived posteriors to the reference posteriors (and vice-versa), aggregated from all voxels, for semisolid MT in mice (f), APT in mice (g), semisolid MT in a subject with glioblastoma (h), and in healthy volunteers (i). The ideal distribution (Mahalanobis distances of samples from any distribution $P$ to same $P$) is shown by the green curve in (f-i), for comparison.