Sean Epstein^{1}, Timothy J.P. Bray^{2}, Margaret A. Hall-Craggs^{2}, and Hui Zhang^{1}

^{1}Centre for Medical Image Computing, University College London, London, United Kingdom, ^{2}University College London, London, United Kingdom

We propose a novel deep learning technique for quantitative MRI parameter estimation. Our method is trained to map noisy qMRI signals to *conventional best-fit parameter labels*, which act as proxies for the groundtruth parameters we wish to estimate. We show that this training leads to more accurate predictions of groundtruth model parameters than traditional approaches which train on these groundtruths directly. Furthermore, we show that our proposed method is both conceptually and empirically equivalent to existing *unsupervised*approaches, with the advantage of being formulated as *supervised* training.

In contrast, our proposed method (

Unsupervised methods implicitly learn this same approximate target function: outputs (parameters), converted to noise-free signal predictions by a qMRI model, are matched to inputs (noisy signals), just as in conventional MLE fitting:

By explicitly learning this approximate target function, our proposed method reformulates unsupervised parameter estimation as a supervised learning problem.

Two signal models were investigated, one non-linear (monoexponential) and one linear (straight line). Model 1 (monoexponential) was selected for (a) its relevance to qMRI (e.g. diffusion, relaxometry) and (b) its non-degeneracy (c.f. multiexponential models), simplifying the interpretation of network performance. Model 2 (straight line) was selected to remove model non-linearity as a potential explanation for our findings.

Simulated datasets were generated at high and low SNRs by generating 100,000 noise-free signals from uniform parameter distributions ($$$\theta_0,\theta_1\in[0.8,1.2]$$$), sampling them ($$$x=[0,0,0,0,1,1,1,1]$$$), and adding noise. Rician noise was added to Model 1 to mimic MRI acquisitions; Gaussian noise was added to Model 2 to remove noise complexity as a potential explanation for our findings. In both cases, low SNR was defined as $$$\sigma=0.10$$$, and high SNR as $$$\sigma=0.05$$$, where $$$max(S_{noisefree})\in[0.8,1.2]$$$.

Network performance was tested on independently generated datasets with sampling, parameter distributions, and noise matching training.

Network architecture was harmonised across all experiments and represents common choices in the existing qMRI literature: 3 fully connected hidden layers, each with 8 nodes, and an output layer of 2 nodes representing the model parameter estimates. Wider networks were investigated and found to have equivalent performance at the cost of increased training time.

As discussed in

Our proposed method brings these benefits of unsupervised networks into a supervised framework by training on independently computed MLE labels. Compared to unsupervised methods, this has the advantage of (a) offloading MLE estimation to separate bespoke algorithms, (b) tailoring training performance to model parameters of interest, and (c) allowing the inclusion of additional information via training labels.

Furthermore, this work sheds light on the relationship between supervised and unsupervised approaches to parameter estimation, and provides a conceptual link between these seemingly-distinct approaches.

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Comparison of our proposed training method (*Supervised*_{MLE}) with traditional supervised (*Supervised*_{GT}) and unsupervised approaches. In all cases, noise-free signals (black) are corrupted with noise (green) and fed as input into the networks. Dashed arrows (red) represent training loss: either between network output and independently-computed MLE estimates (*Supervised*_{MLE}); between network output and groundtruth parameters (*Supervised*_{GT}), or between a noise-free representation of network output and noisy input (unsupervised).

Parameter estimation bias for Model 1 (monoexponential) at low SNR (top) and high SNR (bottom). Arrows point from groundtruth parameter values to mean parameter estimates, averaged over 1000 noise instantiations.

Parameter estimation standard deviation for Model 1 (monoexponential) at low SNR (top) and high SNR (bottom). Each panel shows the mean standard deviation of a given model parameter, at each point in two-dimensional parameter space, averaged over 1000 noise instantiations.

Parameter estimation bias for Model 2 (straight line) at low SNR (top) and high SNR (bottom). Arrows point from groundtruth parameter values to mean parameter estimates, averaged over 1000 noise instantiations.

Parameter estimation standard deviation for Model 2 (straight line) at low SNR (top) and high SNR (bottom). Each panel shows the mean standard deviation of a given model parameter, at each point in two-dimensional parameter space, averaged over 1000 noise instantiations.

DOI: https://doi.org/10.58530/2022/2530