Characterization of anomalous relaxation using the time-fractional Bloch equation and multiple echo T2 *-weighted magnetic resonance imaging at 7T

PURPOSE To study the utility of fractional calculus in modeling gradient-recalled echo MRI signal decay in the normal human brain. METHODS We solved analytically the extended time-fractional Bloch equations resulting in five model parameters, namely, the amplitude, relaxation rate, order of the time-fractional derivative, frequency shift, and constant offset. Voxel-level temporal fitting of the MRI signal was performed using the classical monoexponential model, a previously developed anomalous relaxation model, and using our extended time-fractional relaxation model. Nine brain regions segmented from multiple echo gradient-recalled echo 7 Tesla MRI data acquired from five participants were then used to investigate the characteristics of the extended time-fractional model parameters. RESULTS We found that the extended time-fractional model is able to fit the experimental data with smaller mean squared error than the classical monoexponential relaxation model and the anomalous relaxation model, which do not account for frequency shift. CONCLUSIONS We were able to fit multiple echo time MRI data with high accuracy using the developed model. Parameters of the model likely capture information on microstructural and susceptibility-induced changes in the human brain.

Geometry-free stochastic analysis of BDS triple frequency signals

The paper presents a geometry-free approach to assess the variation of covariance matrices of undifferenced triple frequency GNSS measurements and its impact on positioning solutions. Four independent geometryfree/ ionosphere-free (GFIF) models formed from original triple-frequency code and phase signals allow for effective computation of variance-covariance matrices using real data. Variance Component Estimation (VCE) algorithms are implemented to obtain the covariance matrices for three pseudorange and three carrier-phase signals epoch-by-epoch. Covariance results from the triple frequency Beidou System (BDS) and GPS data sets demonstrate that the estimated standard deviation varies in consistence with the amplitude of actual GFIF error time series. The single point positioning (SPP) results from BDS ionosphere-free measurements at four MGEX stations demonstrate an improvement of up to about 50% in Up direction relative to the results based on a mean square statistics. Additionally, a more extensive SPP analysis at 95 global MGEX stations based on GPS ionosphere-free measurements shows an average improvement of about 10% relative to the traditional results. This finding provides a preliminary confirmation that adequate consideration of the variation of covariance leads to the improvement of GNSS state solutions.