Localized Regularization for Solving the Fredholm Equation of the First Kind with Application to Brain Myelin Mapping

Magnetic resonance relaxometry (MRR) studies of the brain are used to map the distribution of myelin, a complex lipid that forms an insulating sheath around axons and potentiates normative electrical transmission. Measurement of the distribution function (DF) of transverse relaxation times, or T2, can be used to quantify the myelin water fraction (MWF). This involves solving the Fredholm integral equation of the first kind with a Laplace kernel, a notoriously ill-posed problem, with solutions very sensitive to noise. Tikhonov regularization is used to stabilize the determination of the T2 DF, with an optimal regularization parameter, λ, selected according to one of several methods. However, this choice generally represents a compromise, with certain regions of the DF being over-regularized and others being under-regularized. To address this major shortcoming, we present a novel parameter selection method, named localized regularization (LocRegu), that iteratively tunes a vector of λ values across the entire DF, so that each part of the DF may be optimally regularized. We demonstrate the superior performance of LocRegu in simulated data, classical inverse problems, and experimental MRR data for brain MWF determination.