Resolving quantitative MRI model degeneracy in self-supervised machine learning

TL;DR

This work tackles model degeneracy in quantitative MRI (qMRI), where different tissue configurations can produce indistinguishable signals, challenging self-supervised parameter estimation. It shows that self-supervised methods can converge to arbitrary degenerate solutions unless degeneracy is mitigated. The authors propose a degeneracy-resolving framework that constrains the bottleneck output of a physics-based autoencoder and employs a dual-network setup to partition the parameter space into water-dominant and fat-dominant sub-spaces, separated by a boundary at $b=0.58$. Demonstrated on chemical shift-encoded MRI (CSE-MRI) for proton density fat fraction (PDFF) and $R_2^*$ estimation, the approach is validated in silico and in vivo, showing reduced degeneracy and improved alignment with pseudo-ground truths compared with naive self-supervised or supervised methods.

Abstract

Quantitative MRI (qMRI) estimates tissue properties of interest from measured MRI signals. This process is conventionally achieved by model fitting, whose computational expense limits qMRI's clinical use, motivating recent development of machine learning-based methods. Self-supervised approaches are particularly popular as they avoid the pitfall of distributional shift that affects supervised methods. However, it is unknown how such methods behave if similar signals can result from multiple tissue properties, a common challenge known as model degeneracy. Understanding this is crucial for ascertaining the scope within which self-supervised approaches may be applied. To this end, this work makes two contributions. First, we demonstrate that model degeneracy compromises self-supervised approaches, motivating the development of mitigation strategies. Second, we propose a mitigation strategy based on applying appropriate constraining transforms on the output of the bottleneck layer of the autoencoder network typically employed in self-supervised approaches. We illustrate both contributions using the estimation of proton density fat fraction and $R_2^*$ from chemical shift-encoded MRI, an ideal exemplar due to its exhibition of degeneracy across the full parameter space. The results from both simulation and $\textit{in vivo}$ experiments demonstrate that the proposed strategy helps resolve model degeneracy.

Figures (6)

• Figure 1: Diagram of a physics-based autoencoder. A measured signal $\boldsymbol{S}$ is fed to a neural network which predicts tissue properties $\boldsymbol{\Tilde{y}}$. These are then used to reconstruct the signal $\boldsymbol{\Tilde{S}}$ using the forward model, which is used to calculate the self-supervised training loss.
• Figure 2: Rician log-likelihood as a function of PDFF and $R_2^*$ for different tissue parameter combinations. Ground truths are indicated as dotted black lines, representing $R_2^* = 0.1$ ms$^{-1}$ and PDFF equal to 0.2 (left), 0.5 (centre) 0.8 (right). There are two maxima in likelihood, occurring above or below the red dotted switching line at PDFF = 0.58, giving rise to degeneracy.
• Figure 3: Inference procedure using dual network approach. The measured signal is fed to the 'water' and 'fat' networks that make predictions in their respective parameter sub-space via the use of constraining transforms. The final parameter estimates are selected by comparing their likelihood.
• Figure 4: Feasibility of self-supervised learning in the presence of model degeneracy in silico. Without the proposed constraining transforms, self-supervised networks can unpredictably learn to output either water- ($1^{\text{st}}$ column) or fat-dominant ($2^{\text{nd}}$ column) solutions. With the constraining transforms, the target networks are predictably learned ($3^{\text{th}}$ and $4^{\text{th}}$ columns). The prediction that maximises likelihood is chosen as the estimate ($5^{\text{th}}$ column). Rows sequentially represent predicted PDFF values, bias and standard deviation.
• Figure 5: Degeneracy mitigation in vivo using a dual network approach. A single network (leftmost) unpredictably output tissue properties in one degenerate sub-space. The proposed use of constraining transforms ensures 'water' and 'fat' networks are learned with certainty (middle two). The prediction that maximises likelihood is chosen as the estimate (rightmost).