Parameter Estimation in Two-Dimensional Biexponential Magnetic Resonance Relaxometry: A Case-Study Comparison of Frequentist and Bayesian Approaches

In this paper, we contrast frequentist and Bayesian approaches to parameter estimation for a magnetic resonance (MR) relaxometry signal model which takes the form of a two-dimensional biexponential decay. The signal consists of two terms, each parameterized by an amplitude and a transverse and longitudinal relaxation time constant. There are two user-selected parameters, defining the two-dimensional character of the signal; these are an inversion time TI and a set of echo times, TE. Of particular interest is the fact that for two values of TI, which we call the null points, the signal becomes a monoexponential function in TE. Extracting the two parameters—the amplitude and decay constant—from the signal observed at or near a null point is particularly ill-posed since the monoexponential signal is highly overparameterized by the four parameter biexponential models. We seek to estimate these null points, which directly provide values for the longitudinal relaxation time constants, using both frequentist and Bayesian techniques. The frequentist approach uses nonlinear least squares (NLLS), and the Bayesian approach uses the Metropolis–Hastings algorithm. In addition to point estimates, both methods generate point clouds of parameter estimates representing uncertainties. Due to the symmetry of the biexponential model, these point clouds consist of two clusters. The variance of a single cluster and the separation between the two clusters, both of which capture the size of the point clouds, may be used as metrics for ill-posedness. Increasing point cloud size, indicating an undesired greater flexibility in parameter choice, illustrates a greater degree of ill-posedness. We find that both the frequentist and Bayesian approaches can estimate the null points using the extrema of these metrics and yield qualitatively similar and consistent results.