Fractional Order Magnetic Resonance Fingerprinting in the Human Cerebral Cortex

Mathematical models are becoming increasingly important in magnetic resonance imaging (MRI), as they provide a mechanistic approach for making a link between tissue microstructure and signals acquired using the medical imaging instrument. The Bloch equations, which describes spin and relaxation in a magnetic field, are a set of integer order differential equations with a solution exhibiting mono-exponential behaviour in time. Parameters of the model may be estimated using a non-linear solver or by creating a dictionary of model parameters from which MRI signals are simulated and then matched with experiment. We have previously shown the potential efficacy of a magnetic resonance fingerprinting (MRF) approach, i.e., dictionary matching based on the classical Bloch equations for parcellating the human cerebral cortex. However, this classical model is unable to describe in full the mm-scale MRI signal generated based on an heterogenous and complex tissue micro-environment. The time-fractional order Bloch equations have been shown to provide, as a function of time, a good fit of brain MRI signals. The time-fractional model has solutions in the form of Mittag-Leffler functions that generalise conventional exponential relaxation. Such functions have been shown by others to be useful for describing dielectric and viscoelastic relaxation in complex heterogeneous materials. Hence, we replaced the integer order Bloch equations with the previously reported time-fractional counterpart within the MRF framework and performed experiments to parcellate human gray matter, which consists of cortical brain tissue with different cyto-architecture at different spatial locations. Our findings suggest that the time-fractional order parameters, alpha and beta, potentially associate with the effect of interareal architectonic variability, which hypothetically results in more accurate cortical parcellation.

Automated summary

Mathematical models play a crucial role in linking tissue microstructure with MRI signals. The Bloch equations describe spin and relaxation in a magnetic field, while the time-fractional order Bloch equations provide a better fit for brain MRI signals. Time-fractional models offer solutions in the form of Mittag-Leffler functions, which have been proven useful for describing relaxation in complex materials.