Magnetic resonance described in the excitation dependent rotating frame of reference.

An excitation dependent rotating frame of reference to observe the magnetic resonance phenomenon is introduced in this paper that, to the best of our knowledge, has not been used previously in the nuclear magnetic resonance context. The mathematical framework for this new rotating frame of reference is presented based on time scaling the Bloch equation after transformation to the classical rotating frame of reference whose transverse plane is rotating at the Larmor frequency. To this end, the Bloch equation is rewritten in terms of a magnetisation vector observed from the excitation dependent rotating frame of reference. The resultant Bloch equation is referred to as the time scaled Bloch equation. In the excitation dependent rotating frame of reference whose coordinates are rotating at the instantaneous Rabi frequency the observed magnetisation vector is a much slower signal than the true magnetisation in the rotating frame of reference. As a result the ordinary differential equation solvers have the ability to solve the time scaled version of the Bloch equation with a larger step size resulting in a smaller number of samples for solving the equation to a desired level of accuracy. The simulation results for different types of excitation are presented in this paper. This method may be used in true Bloch simulators in order to reduce the simulation time or increase the accuracy of the numerical solution. Moreover, the time scaled Bloch equation may be employed to determine the optimal excitation pattern in magnetic resonance imaging as well as designing pulses with better slice selectivity which is an active area of research in this field.