Physics-Informed Sylvester Normalizing Flows for Bayesian Inference in Magnetic Resonance Spectroscopy

TL;DR

The paper addresses the challenging problem of quantifying metabolites from proton MRS in the presence of spectral overlap and noise by introducing a Bayesian framework that leverages Sylvester normalizing flows to approximate the posterior distribution over metabolite concentrations and spectral parameters. A physics-informed forward model decodes latent variables into realistic spectra, enabling calibrated uncertainty estimates and interpretation of parameter correlations. The authors demonstrate improved posterior quality and uncertainty calibration on simulated 7T MRS data compared to LCModel and a non-flow VAE, with evidence of multi-modal posteriors and metabolite dependencies. This approach offers a principled, interpretable pathway toward more reliable metabolite quantification in clinical MRS workflows.

Abstract

Magnetic resonance spectroscopy (MRS) is a non-invasive technique to measure the metabolic composition of tissues, offering valuable insights into neurological disorders, tumor detection, and other metabolic dysfunctions. However, accurate metabolite quantification is hindered by challenges such as spectral overlap, low signal-to-noise ratio, and various artifacts. Traditional methods like linear-combination modeling are susceptible to ambiguities and commonly only provide a theoretical lower bound on estimation accuracy in the form of the Cramér-Rao bound. This work introduces a Bayesian inference framework using Sylvester normalizing flows (SNFs) to approximate posterior distributions over metabolite concentrations, enhancing quantification reliability. A physics-based decoder incorporates prior knowledge of MRS signal formation, ensuring realistic distribution representations. We validate the method on simulated 7T proton MRS data, demonstrating accurate metabolite quantification, well-calibrated uncertainties, and insights into parameter correlations and multi-modal distributions.

Figures (5)

• Figure 1: Illustration of key challenges in mrs metabolite quantification. From left to right: a noiseless spectrum with individual metabolite components, showing substantial spectral overlap; the same spectrum with broadened peaks due to poor B0 shimming, increasing overlap; and a low-snr spectrum (snr = 7 dB), where noise obscures low-intensity metabolites.
• Figure 2: The inference network takes the observed spectrum $x$ as input and outputs the parameters of the base distribution $q_0(\theta_0 \,|\,x)$ for the latent variables $\theta$. snf are then applied to the base distribution to transform it into a more complex and flexible posterior distribution $q_K(\theta_K \,|\,x)$. Instead of a decoder network, the physics-based mrs signal model directly maps the latent variables $\theta_K$, representing metabolite concentrations and other relevant parameters, to the spectral domain.
• Figure 3: Example test spectrum $\hat{x}$ fit using the snf and LCModel. Top left: reconstructed spectrum obtained by sampling from the snf posterior. Top right: LCModel fit for the same spectrum. Bottom: pairplot of snf-inferred posterior distributions for cr, gaba, glu, gpc, naa, pch, and pcr. Marginal distributions are shown along the diagonal; correlations are shown off-diagonal. Prior simulation ranges are indicated by dashed boxes; LCModel estimates and crlb are overlaid for reference.
• Figure 4: mae and uncertainty estimates for cr, gaba, glu, and mins across varying linewidths, evaluated on 1000 test spectra. The standard deviation is estimated from 1000 posterior samples for snf and vae, and from the crlb for LCModel.
• Figure 5: Calibration curves for snf, vae, and LCModel, depicting nominal versus empirical coverage for each metabolite across 1000 test spectra. The ideal calibration is shown as a dashed line, with nominal coverage ranging from 1% to 99%.