Purpose: Parameter estimation from biexponential models is a notoriously ill-posed problem, often yielding inaccurate estimates that are strongly influenced by noise. We present a simple, nonconventional approach to regularizing nonlinear least-squares (NLLS) parameter estimation and apply it to myelin water fraction (MWF) mapping with magnetic resonance imaging. Motivated by the ridge regression theorem for linear least-squares problems, we introduce a nonlinear extension, which we denote $\lambda$-NL-RR, in which regularization is applied directly to the NLLS estimation of discrete parameters. Methods: We applied $\lambda$-NL-RR to simulated and experimental biexponential magnetic resonance imaging (MRI) data and compared parameter estimates with those obtained using conventional NLLS fitting. The regularization parameter $\lambda$ was selected using generalized cross validation. Results: Under conditions of modest signal-to-noise ratio and closely spaced exponential time constants, regimes in which conventional biexponential analysis performs particularly poorly, $\lambda$-NL-RR was found to reduce mean-square error by up to 80% relative to standard NLLS for simulated data, consistent with the known behavior of regularization in more conventional contexts. Improvements in root-mean-square error in the range of 11%–25% in parameter estimates were seen for in vivo brain data, although there is no true gold standard for underlying values in these experiments. Notably, these results are due to postprocessing only and require no modification of data acquisition strategy or duration. Conclusion: Regularization of the NLLS estimation problem for the biexponential model substantially improves parameter accuracy in MWF mapping. Although related ideas exist in scattered form in the applied mathematics literature, this work recognizes, formalizes, and demonstrates ridge-type regularization for discrete NLLS parameter estimation in MRI. These results establish a foundation for broader application of nonlinear regularization strategies in quantitative MRI and other fields.

Close

PrevNext

View figure in article

• Inverse problems
• Medical magnetic resonance imaging
• Biological materials
• Brain
• Tissues
• Water
• Data analysis
• Statistical methods