Input layer regularization and automated regularization hyperparameter tuning for myelin water estimation using deep learning

TL;DR

This work tackles the ill-posed problem of estimating the myelin water fraction (MWF) from biexponential MR relaxometry signals by integrating traditional regularization with deep learning through Input Layer Regularization (ILR). The authors develop two pipelines—one using neural network-based lambda selection ($ ext{lambda}_{ ext{NN}}$) and one using generalized cross validation ($ ext{lambda}_{ ext{GCV}}$)—and augment the input to a parameter-estimation network with a regularized signal derived from the current lambda estimate, forming the augmented input $ extbf{x}=[ extbf{s}, extbf{G}( extbf{p}^*_{ ext{lambda}}( extbf{s}))]$. They extend ILR to the three-parameter biexponential model and demonstrate improved accuracy in estimating $c_1$ (MWF fraction) on synthetic data and in vivo brain data, with GCV offering robustness at low SNR and the NN offering strong performance at higher SNR. The study shows that hybrid classical-ML approaches can outperform purely data-driven methods in medical imaging tasks and provides a practical framework for automated regularization parameter tuning in MR relaxometry.

Abstract

We propose a novel deep learning method which combines classical regularization with data augmentation for estimating myelin water fraction (MWF) in the brain via biexponential analysis. Our aim is to design an accurate deep learning technique for analysis of signals arising in magnetic resonance relaxometry. In particular, we study the biexponential model, one of the signal models used for MWF estimation. We greatly extend our previous work on \emph{input layer regularization (ILR)} in several ways. We now incorporate optimal regularization parameter selection via a dedicated neural network or generalized cross validation (GCV) on a signal-by-signal, or pixel-by-pixel, basis to form the augmented input signal, and now incorporate estimation of MWF, rather than just exponential time constants, into the analysis. On synthetically generated data, our proposed deep learning architecture outperformed both classical methods and a conventional multi-layer perceptron. On in vivo brain data, our architecture again outperformed other comparison methods, with GCV proving to be somewhat superior to a NN for regularization parameter selection. Thus, ILR improves estimation of MWF within the biexponential model. In addition, classical methods such as GCV may be combined with deep learning to optimize MWF imaging in the human brain.

Figures (10)

• Figure 1: The (ND, Reg) parameter estimation network is an NN where the inputs are the concatenated vectors of noisy signal $s$ and the signal obtained from parameters estimated by Tikhonov regularization, with regularization parameter $\lambda(\textbf{s})$ given by either a neural network or GCV. The parameters estimated are $c_1, T_{2,1}$ and $T_{2,2}$.
• Figure 2: Combination of signal-dependent $\lambda$ selection with input layer regularization. The first step involves selecting either a NN or GCV for estimating $\lambda$. Training data for the ILR networks $\textsc{(ND, Reg)}$ are first passed through a TR-NLLS solver with the signal-dependent $\lambda(\textbf{s})$ to construct the concatenated vector $\textbf{x}$ in Eq. \ref{['eq: concatenated vector']}. $C(\lambda)$ is defined according to Eq. \ref{['eq: GCV function']}
• Figure 3: $\lambda_{NN}$ architecture. 1-dimensional convolutional layers are applied to the input noisy signal, followed by fully connected layers to predict $\lambda_{oracle}$. $C$ refers to the channel sizes (in/out for input or output channels, respectively). We also define the kernel size as $K_{size}$ and the padding as $P_{size}$. $W$ refers to the width of a given fully-connected layer. The ReLU activation function is used between each layer.
• Figure 4: 3D Histogram showing prediction density of the $\lambda$-selection neural network versus the $\lambda_{\text{oracle}}$ approach detailed in Algorithm \ref{['alg:datasetGen']}. The white line represents the $y=x$ line, indicating the benchmark for perfect prediction. Left to right: SNRs 5 (low), 50 (medium), 100 (high).
• Figure 5: Left to right: The true $\lambda$ distribution (orange) overlaid with the distributions of $\lambda_{NN}$ and $\lambda_{GCV}$. The true distributions at the low SNR level are more multimodal and have greater spread, with GCV providing superior estimates as compared to NN $\lambda$ selection. This is particularly seen at lower values of $\lambda$, where little to no regularization is needed. At the higher SNR's, the optimal $\lambda$ values are concentrated around a small number of modes, with approximation of $\lambda_{oracle}$ by the NN now exhibiting performance superior to GCV.