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Purpose MRI relaxation measurements are performed in the presence of a fictitious magnetic field in the recently described technique known as RAFF (Relaxation Along a Fictitious Field). This method operates in the 2nd rotating frame (rank n = 2) by utilizing a non-adiabatic sweep of the radiofrequency effective field to generate the fictitious magnetic field. In the present study, the RAFF method is extended for generating MRI contrasts in rotating frames of ranks 1 ≤ n ≤ 5. The developed method is entitled RAFF in rotating frame of rank n (RAFFn). Methods RAFFn pulses were designed to generate fictitious fields that allow locking of magnetization in rotating frames of rank n. Contrast generated with RAFFn was studied using Bloch-McConnell formalism together with experiments on human and rat brains. Results Tolerance to B0 and B1 inhomogeneities and reduced specific absorption rate with increasing n in RAFFn were demonstrated. Simulations of exchange-induced relaxations revealed enhanced sensitivity of RAFFn to slow exchange. Consistent with such feature, an increased grey/white matter contrast was observed in human and rat brain as n increased. Conclusion RAFFn is a robust and safe rotating frame relaxation method to access slow molecular motions in vivo.

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Nonstationary inverse problems are usually cast in the state-space formalism. The complete statistics of linear Gaussian problems can be computed with the Kalman filters and smoothers. Nonlinear non-Gaussian problems would necessitate the adoption of particle filters or similar computationally very heavy approaches. The so-called extended Kalman filters often provide suboptimal but feasible estimates for the nonlinear problems. Several applications which lead to nonstationary inverse problems are time critical, such as process tomography and many biomedical problems. In such applications, there is typically a need to use reduced-order models which may heavily compromise the computational accuracy that is usually required for inverse problems. One approach to overcome the model reduction problem is to use the approximation error analysis. In this paper, we derive the equations for the extended Kalman filter for nonlinear state estimation problems in which the approximation error models are taken into account. We consider the approximation errors that are due to both state reduction and time stepping. As an example, we consider the identification of the coefficients of the heat equation. Our main result is that, also in nonlinear problems, approximation error analysis allows us to obtain accurate estimates and uncertainties of the parameters in reduced-order models that are suitable for fast calculation.