Myelin Mapping in the Human Brain Using an Empirical Extension of the Ridge Regression Theorem

Myelin water fraction (MWF) mapping in the central nervous system is a topic of intense research activity. One framework for this requires parameter estimation from a decaying biexponential signal. However, this is often an ill-posed nonlinear problem resulting in unreliable parameter estimates. For linear least-squares (LLS) problems, the ridge regression theorem (RRT) shows that a Tikhonov regularization parameter exists that will reduce mean square error (MSE) in parameter estimates. We present and apply a nonlinear version of the RRT, λ -NL-RR, to MWF mapping.

Methods

For simulated and experimental data, we estimated parameter values with conventional nonlinear least-squares (NLLS) and compared these with values obtained from λ -NL-RR, with the regularization parameter value defined by generalized cross validation. We applied regularization only to signals identified as biexponential according to the Bayesian information criterion.

Results

Under conditions of modest SNR and closely spaced exponential time constants in which conventional biexponential analysis methods yield particularly inaccurate results, λ -NL-RR decreases MSE by ~10-15%.

Conclusion

Regularization of the NLLS parameter estimation problem for the biexponential model decreased MSE for simulated and in vivo MRI brain data. In addition, this work provides a general framework for regularization of a broad class of NLLS problems.