Condition Number Phase Transitions for Model Selection in Magnetic Resonance Relaxometry

In magnetic resonance (MR), the biexponential model describes the decay signal of macromolecules in ambient non-bound water. This provides an important means of assessing myelin status in the brain. The inversion-recovery experiment in MR, perhaps uniquely in experimental science, permits the nulling of a selected term in a biexponential model. At the inversion times (TI) resulting in component nulling, experimental data is monoexponential, so that fitting to the full biexponential model is underdetermined and results in parameter instability. We use this fact to identify the null points numerically by observing the standard deviation (SD), across noise realizations, of parameter estimates as a function of TI. Knowledge of these values of TI reduces the dimensionality of the parameter estimation problem. The Bayesian information criterion provides a further indication of the null points by identifying the range of TI values at which the signal is better-fitted with a mono- rather than a biexponential model. Finally, we use persistence homology from topological data analysis to characterize the range of acceptable parameter estimates to determine the null points. These methods all permit improved stability of parameter estimates for the two-dimensional biexponential model in MR.