Bias-Reduced Neural Networks for Parameter Estimation in Quantitative MRI

TL;DR

This work tackles bias and variance in neural-network estimators for quantitative MRI by generalizing the training loss to average over noise realizations and enforce efficiency-like properties through CRB-weighted, variance-constrained terms. The proposed Bias-Reduced loss reduces estimator bias across parameter space while maintaining variance near the Cramér-Rao bound, yielding results that closely match traditional estimators in vivo but with far greater computational efficiency. Applied to magnetization transfer and MR fingerprinting tasks, the approach demonstrates improved bias control without sacrificing accuracy, offering a practical path toward robust, automated qMRI biomarker estimation. It also discusses limitations such as data mismatch and suggests extensions to incorporate richer covariance structures and uncertainty quantification.

Abstract

Purpose: To develop neural network (NN)-based quantitative MRI parameter estimators with minimal bias and a variance close to the Cramér-Rao bound. Theory and Methods: We generalize the mean squared error loss to control the bias and variance of the NN's estimates, which involves averaging over multiple noise realizations of the same measurements during training. Bias and variance properties of the resulting NNs are studied for two neuroimaging applications. Results: In simulations, the proposed strategy reduces the estimates' bias throughout parameter space and achieves a variance close to the Cramér-Rao bound. In vivo, we observe good concordance between parameter maps estimated with the proposed NNs and traditional estimators, such as non-linear least-squares fitting, while state-of-the-art NNs show larger deviations. Conclusion: The proposed NNs have greatly reduced bias compared to those trained using the mean squared error and offer significantly improved computational efficiency over traditional estimators with comparable or better accuracy.

Figures (5)

• Figure 1: Boxplot comparison of simulated qMT parameter fits with networks trained using a state-of-the-art (MSE-CRB or Cramér-Rao bound weighted mean squared error criterionZhang2022) and the proposed Bias-Reduced loss, assuming SNR $=20$. As an example, we vary only $R_2^\mathrm{f}$ (the free spin pool's transverse relaxation rate) while keeping all other parameters constant (red reference lines are the ground-truth). The proposed strategy significantly reduces the variable bias in all other parameters throughout parameter space except $T_2^\mathrm{s}$ ($p<10^{-3}$ using Welch's unequal variances $t$-test).
• Figure 2: Normalized bias and standard deviation of the magnetization transfer parameter estimates as a function of SNR ($|M_0|/\sigma$). The estimates are based on simulations using typical white matter values ($m_0^\mathrm{s}=0.2$, $R_1^\mathrm{f}=0.52/s$, $R_2^\mathrm{f}=12.9/s$, $R_\mathrm{x}=16.5/s$, $R_1^\mathrm{s}=2.97/s$, $T_2^\mathrm{s}=12.4\mu s$),Asslander2023 which are used for normalization. We compare neural networks trained using the MSE-CRBZhang2022 and proposed Bias-Reduced ($\lambda=1$) losses to non-linear least squares (NLLS) and a hypothetical efficient estimator, which has zero bias and variance equal to the Cramér-Rao bound. The proposed strategy is similar in performance to NLLS in all parameters except $T_2^\mathrm{s}$, where it more closely matches an efficient estimator. This analysis, repeated for grey matter values, is shown in Sup. Fig. \ref{['sfig:mt_bv']}.
• Figure 3: Normalized histograms of the CRB-weighted ($\mathbf{b}_{ik}$) squared bias (A--F) and variance (G--L) of the magnetization transfer parameter estimates $\{\hat{\mathbf{x}}_{ik}\}$, where $i$ indexes across the test set (where each fingerprint has a random SNR) and $k$ indexes across parameters. Note the scaling of the y-axis truncates the left-most bins in each subplot. The MSE-CRBZhang2022 loss (blue) offers the lowest variance but the highest bias overall in comparison to the proposed Bias-Reduced strategies $\lambda=1$ (red) and $\lambda=0.1$ (green). A smaller $\lambda$ ($\lambda=0.1$) reduces the overall bias slightly at the outsized cost of an increased proportion of fingerprints exceeding the CRB ($\delta=1$; Eq. \ref{['eq:wvce']}). A comparison to Eq. \ref{['eq:wbce']} is shown in Sup. Fig. \ref{['sfig:mt_hist']}.
• Figure 4: In vivo magnetization transfer parameter maps fitted with the MSE-CRBZhang2022 and proposed Bias-Reduced neural networks in comparison to a non-linear least squares (NLLS) reference. The Bias-Reduced network offers the highest visual contrast in $m_0^\mathrm{s}$ (magnifications) and has improved consistency with NLLS in all parameters (particularly $R_1^\mathrm{f}$ and $R_\mathrm{x}$, red arrows) except for $T_2^\mathrm{s}$, consistent with Fig. \ref{['fig:mt_bv']}F.
• Figure 5: In vivo $1/T_1$ (A--C) and $1/T_2$ (D--F) maps acquired using the MRF-FISP sequence and fitted using NNs trained with two different strategies in comparison to a dictionary-matching-based reference. (G) analyzes the $T_2$ values within the two white matter ROIs drawn in (F), where outliers are not plotted. The Bias-Reduced NN yields more similar parameter maps to those of dictionary matching, but with the benefit of improved computational efficiency. Similar accuracy and precision to dictionary matching is also observed in simulation (Sup. Figs. \ref{['sfig:fisp_bv_t2']}--\ref{['sfig:fisp_bv_t1']}).