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Published on: October 24, 2012
Myelin Mapping in the Human Brain Using an Empirical Extension of the Ridge Regression Theorem
Purpose:
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.
