Use of Nonlinear Ridge Regression for Discrete Parameter Estimation in Magnetic Resonance Models

The use of magnetic resonance (MR) has been increasingly directed towards quantitative studies, which involve fitting MR imaging data to mathematical models. Examples include myelin water fraction (MWF) mapping, diffusion kurtosis, and calf muscle bioenergetics. These examples are inverse problems, meaning that parameters are extracted by fitting data to a model equation.

A common approach to solving inverse problems is nonlinear least squares (NLLS), in which parameters are estimated by minimizing the residual sum of squares (RSS) between experimental data and a model equation. Unfortunately, different parameter sets can produce very similar-looking curves, demonstrating that minimizing RSS is insufficient for accurate parameter estimation.

Ridge regression (RR) is an unconventional method that can improve upon parameter estimates by adding a shrinkage penalty to NLLS. There is limited literature on which method is the best for selecting the RR parameter, λ. Here, we apply RR to improve parameter estimation across several biological MR models, and evaluate multiple methods of λ selection.

Our results indicate that λ selection through a feed-forward neural network and generalized cross-validation (GCV) can successfully improve parameter estimation of many MR models.